{"id":"n-1e3d729fffbceff3","revision":0,"first_index":0,"fingerprint":"f6-e99c3824c4864868","name":"one_plus_derivative of basis_map_A013610 of Triangle read by rows: T(n,k) is the number of ternary words of length n having k levels (n >= 1, 0 <= k <= n-1). A level is a pair of identical consecutive letters.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["3"],["18","9"],["81","108","27"],["324","729","486","81"],["1215","3888","4374","1944","243"],["4374","18225","29160","21870","7290","729"],["15309","78732","164025","174960","98415","26244","2187"],["52488","321489","826686","1148175","918540","413343","91854","6561"],["177147","1259712","3857868","6613488","6889050","4408992","1653372","314928","19683"],["590490","4782969","17006112","34720812","44641044","37200870","19840464","6377292","1062882","59049"],["1948617","17714700","71744535","170061120","260406090","267846264","186004350","85030560","23914845","3542940","177147"],["6377292","64304361","292292550","789189885","1403004240","1718680194","1473154452","876877650","350751060","87687765","11691702","531441"],["20726199","229582512","1157478498","3507510600","7102708965","10101630528","10312081164","7576222896","3945949425","1403004240","315675954","38263752","1594323"],["66961566","808321761","4476858984","15047220474","34198228350","55401129927","65660598432","57453023628","36934086618","17099114175","5471716536","1119214746","124357194","4782969"],["215233605","2812385772","16974756981","62676025776","157995814977","287265118140","387807909489","393963590592","301628374047","172359070884","71816279535","20892008592","3917251611","401769396","14348907"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["3","27","216","1620","11664","81648","559872","3779136","25194240","166281984","1088391168","7074542592","45712429056","293865615360","1880739938304"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["3"]],"recurrence":{"denominator":{"coeffs":[["0","1"],["1","0"]]},"terms":[{"coeff":{"coeffs":[["3","6","3"],["3","3","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":181791,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-1a75bdb00b3f42e8"]},{"id":181792,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n have one sign","region_exact":null,"edges":["e-1a75bdb00b3f42e8"]},{"id":181793,"position":2,"claim":"hadamard_kernel","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":0},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-1a75bdb00b3f42e8"]},{"id":181794,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-1a75bdb00b3f42e8"]}],"region_exact":{},"incoming_edges":[{"id":"e-1a75bdb00b3f42e8","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-1e3d729fffbceff3","premise_count":1,"theorem":"Laguerre's theorem on differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"19159e6e5bca6d171f54984ed9f2c870bc4e3b75993bd20cd37db722b39d5c25","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-97ce69a237fb4e95","role":"sequence","name":"basis_map_A013610 of Triangle read by rows: T(n,k) is the number of ternary words of length n having k levels (n >= 1, 0 <= k <= n-1). A level is a pair of identical consecutive letters.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-006ac39657b3c661","kind":"fingerprint_related","node_a_id":"n-0040332d7acdddb4","node_b_id":"n-1e3d729fffbceff3","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-0040332d7acdddb4","other":{"id":"n-0040332d7acdddb4","name":"theta_plus_one of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["2"],["2","4"],["2","8","6"]],"card_rows":[["2"],["2","4"],["2","8","6"],["2","12","18","8"]],"row_sums_preview":["2","6","16","40","96","224","512","1152"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-089a9b4f824a1c20","steps":["\u03b8+1"],"edges":["e-421af44bc39d42c8"],"text":"A028326 \u2192 \u03b8+1"},"headline":"A028326 \u2192 \u03b8+1","chain":null}},{"id":"l-0198b239ebce198a","kind":"fingerprint_related","node_a_id":"n-1e3d729fffbceff3","node_b_id":"n-7dacda6114b9f10f","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-7dacda6114b9f10f","other":{"id":"n-7dacda6114b9f10f","name":"one_plus_derivative of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1728","2304","576"],["55296","124416","82944","13824"],["1658880","5308416","5971968","2654208","331776"]],"card_rows":[["1728","2304","576"],["55296","124416","82944","13824"],["1658880","5308416","5971968","2654208","331776"],["47775744","199065600","318504960","238878720","79626240","7962624"]],"row_sums_preview":["4608","276480","15925248","891813888","48922361856","2641807540224","140896402145280","7439330033270784"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c1532953a6736702","steps":["basis A013609","D+1"],"edges":["e-9ec880266e74c89d","e-96acbe56746d19e3"],"text":"A038338 \u2192 basis A013609 \u2192 D+1"},"headline":"A038338 \u2192 basis A013609 \u2192 D+1","chain":null}},{"id":"l-023bd9040bb9bc16","kind":"fingerprint_related","node_a_id":"n-7a1376d2a2b6b46f","node_b_id":"n-1e3d729fffbceff3","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-7a1376d2a2b6b46f","other":{"id":"n-7a1376d2a2b6b46f","name":"one_plus_derivative of Triangle read by rows: T(n,k) = ((n+1)(n+2)/2) * binomial(n,k) (0 <= k <= n).","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6","3"],["18","24","6"]],"card_rows":[["1"],["6","3"],["18","24","6"],["40","90","60","10"]],"row_sums_preview":["1","9","48","200","720","2352","7168","20736"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7f1b09730105c8f0","steps":["D+1"],"edges":["e-1312db2a2710f639"],"text":"A094305 \u2192 D+1"},"headline":"A094305 \u2192 D+1","chain":null}},{"id":"l-052721b3fa4dfe89","kind":"fingerprint_related","node_a_id":"n-ceaac161de738391","node_b_id":"n-1e3d729fffbceff3","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-ceaac161de738391","other":{"id":"n-ceaac161de738391","name":"one_plus_derivative of A Jacobsthal-Pascal triangle.","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","1"],["9","12","3"]],"card_rows":[["1"],["2","1"],["9","12","3"],["20","45","30","5"]],"row_sums_preview":["1","3","24","100","528","2352","11008","48960"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-9f2158ef8512cadd","steps":["D+1"],"edges":["e-4772f2d2b5dd35f9"],"text":"A124860 \u2192 D+1"},"headline":"A124860 \u2192 D+1","chain":null}},{"id":"l-0b2217b0a264e13f","kind":"fingerprint_related","node_a_id":"n-801895951aa20f29","node_b_id":"n-1e3d729fffbceff3","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-801895951aa20f29","other":{"id":"n-801895951aa20f29","name":"Triangle read by rows: n-th row = unsigned coefficients of the characteristic polynomials of an n X n matrix with 2's on the diagonal and 1's elsewhere.","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","2"],["1","4","3"]],"card_rows":[["1"],["1","2"],["1","4","3"],["1","6","9","4"]],"row_sums_preview":["1","3","8","20","48","112","256","576"],"regions":["(-inf, 0]","(-inf, 0)"],"region_exact":{},"oeis_ids":["A103406"],"derivation":{"root":"n-801895951aa20f29","steps":[],"edges":[],"text":"A103406"},"headline":"A103406: Triangle read by rows: n-th row = unsigned coefficients of the characteristic polynomials of an n X n matrix with 2's on the diagonal and 1's elsewhere.","chain":null}},{"id":"l-0fd19751c99e1adc","kind":"fingerprint_related","node_a_id":"n-c4d59c13bd5c1d27","node_b_id":"n-1e3d729fffbceff3","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-c4d59c13bd5c1d27","other":{"id":"n-c4d59c13bd5c1d27","name":"one_plus_derivative of Triangle whose (i,j)-th entry is binomial(i,j)*2^i.","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","2"],["12","16","4"]],"card_rows":[["1"],["4","2"],["12","16","4"],["32","72","48","8"]],"row_sums_preview":["1","6","32","160","768","3584","16384","73728"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d28ad9841fc73317","steps":["D+1"],"edges":["e-8aab31d568b7b201"],"text":"A038208 \u2192 D+1"},"headline":"A038208 \u2192 D+1","chain":null}},{"id":"l-127df65bfc3b597d","kind":"fingerprint_related","node_a_id":"n-1e3d729fffbceff3","node_b_id":"n-a6f98df96412c71b","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-a6f98df96412c71b","other":{"id":"n-a6f98df96412c71b","name":"one_plus_derivative of Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","1"],["6","8","2"]],"card_rows":[["1"],["2","1"],["6","8","2"],["24","54","36","6"]],"row_sums_preview":["1","3","16","120","1152","13440","184320","2903040"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cb5d49e71e46f2ed","steps":["D+1"],"edges":["e-9664c8efaf98340a"],"text":"A196347 \u2192 D+1"},"headline":"A196347 \u2192 D+1","chain":null}},{"id":"l-143970568d818ea8","kind":"fingerprint_related","node_a_id":"n-e9f1e9290023f124","node_b_id":"n-1e3d729fffbceff3","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-e9f1e9290023f124","other":{"id":"n-e9f1e9290023f124","name":"one_plus_derivative of basis_map_A013609 of basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-4","-2"],["12","16","4"]],"card_rows":[["1"],["-4","-2"],["12","16","4"],["-32","-72","-48","-8"]],"row_sums_preview":["1","-6","32","-160","768","-3584","16384","-73728"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-74c1e94669ff21ff","steps":["basis A013609","D+1"],"edges":["e-b6cdc8ab0f2a09ef","e-02dc263b688c42c9"],"text":"basis-map coefficients of A113214, shifted \u2192 basis A013609 \u2192 D+1"},"headline":"basis-map coefficients of A113214, shifted \u2192 basis A013609 \u2192 D+1","chain":null}},{"id":"l-15ef785d320df21e","kind":"fingerprint_related","node_a_id":"n-b93164230fee0958","node_b_id":"n-1e3d729fffbceff3","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-b93164230fee0958","other":{"id":"n-b93164230fee0958","name":"one_plus_derivative of Triangle read by rows: T(n,k) = (2^(n+1)-1)*binomial(n,k).","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6","3"],["21","28","7"]],"card_rows":[["1"],["6","3"],["21","28","7"],["60","135","90","15"]],"row_sums_preview":["1","9","56","300","1488","7056","32512","146880"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8d96419786c3ee2a","steps":["D+1"],"edges":["e-26c90303f05dada8"],"text":"A134346 \u2192 D+1"},"headline":"A134346 \u2192 D+1","chain":null}},{"id":"l-17ea53cf7568db4b","kind":"fingerprint_related","node_a_id":"n-76679d10daed395b","node_b_id":"n-1e3d729fffbceff3","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-76679d10daed395b","other":{"id":"n-76679d10daed395b","name":"one_plus_derivative of shift_x_plus_one of 3n + 1 preceded by n zeros.","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","4"],["21","28","7"]],"card_rows":[["1"],["8","4"],["21","28","7"],["40","90","60","10"]],"row_sums_preview":["1","12","56","200","624","1792","4864","12672"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ccc86c3dcf2ac685","steps":["x \u21a6 x+1","D+1"],"edges":["e-28824f426538156a","e-26afa27b31d8fc25"],"text":"A131431 \u2192 x \u21a6 x+1 \u2192 D+1"},"headline":"A131431 \u2192 x \u21a6 x+1 \u2192 D+1","chain":null}},{"id":"l-193efe174d8c616c","kind":"fingerprint_related","node_a_id":"n-1e3d729fffbceff3","node_b_id":"n-691996f37644fa1b","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-691996f37644fa1b","other":{"id":"n-691996f37644fa1b","name":"theta_plus_one of basis_map_A013609 of Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["12","24"],["84","336","252"]],"card_rows":[["1"],["12","24"],["84","336","252"],["448","2688","4032","1792"]],"row_sums_preview":["1","36","672","8960","96768","903168","7569408","58392576"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-97228a88c81561dc","steps":["basis A013609","\u03b8+1"],"edges":["e-5270021ea4ce6d61","e-aca61d6095d5429b"],"text":"A178822 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A178822 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}},{"id":"l-252eea4c9ba82fcb","kind":"fingerprint_related","node_a_id":"n-1e3d729fffbceff3","node_b_id":"n-38d376fa38bc637e","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-38d376fa38bc637e","other":{"id":"n-38d376fa38bc637e","name":"theta_plus_one of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*3^j.","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6","12"],["36","144","108"]],"card_rows":[["1"],["6","12"],["36","144","108"],["216","1296","1944","864"]],"row_sums_preview":["1","18","288","4320","62208","870912","11943936","161243136"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8d88ef9de74b305f","steps":["basis A013609","\u03b8+1"],"edges":["e-86b1b19edf92ec9a","e-cc5ad339e9f55a99"],"text":"A038221 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A038221 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}},{"id":"l-2565ac8d2a2c04b8","kind":"fingerprint_related","node_a_id":"n-65d62a2c475a7137","node_b_id":"n-1e3d729fffbceff3","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-65d62a2c475a7137","other":{"id":"n-65d62a2c475a7137","name":"theta_plus_one of Triangle read by rows, for 0 <= k <= n: T(n, k) = binomial(n, k) * binomial(binomial(n + 3, 2), 2).","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["3"],["15","30"],["45","180","135"]],"card_rows":[["3"],["15","30"],["45","180","135"],["105","630","945","420"]],"row_sums_preview":["3","45","360","2100","10080","42336","161280","570240"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-dd4aa31f3d11d152","steps":["\u03b8+1"],"edges":["e-0a30318f64ff8ab1"],"text":"A344565 \u2192 \u03b8+1"},"headline":"A344565 \u2192 \u03b8+1","chain":null}},{"id":"l-280022131417ff59","kind":"fingerprint_related","node_a_id":"n-1e3d729fffbceff3","node_b_id":"n-13e35bca57344dde","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-13e35bca57344dde","other":{"id":"n-13e35bca57344dde","name":"theta_plus_one of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["256","1024","768"],["4096","24576","36864","16384"],["65536","524288","1179648","1048576","327680"]],"card_rows":[["256","1024","768"],["4096","24576","36864","16384"],["65536","524288","1179648","1048576","327680"],["1048576","10485760","31457280","41943040","26214400","6291456"]],"row_sums_preview":["2048","81920","3145728","117440512","4294967296","154618822656","5497558138880","193514046488576"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a93042cd70c4761e","steps":["basis A013611","\u03b8+1"],"edges":["e-dab838a9194ad2c3","e-8faa2a8c53b3db32"],"text":"A038330 \u2192 basis A013611 \u2192 \u03b8+1"},"headline":"A038330 \u2192 basis A013611 \u2192 \u03b8+1","chain":null}},{"id":"l-28a5892ed8f320c8","kind":"fingerprint_related","node_a_id":"n-1e3d729fffbceff3","node_b_id":"n-f6567e201d984692","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-f6567e201d984692","other":{"id":"n-f6567e201d984692","name":"one_plus_derivative of basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["768","1024","256"],["16384","36864","24576","4096"],["327680","1048576","1179648","524288","65536"]],"card_rows":[["768","1024","256"],["16384","36864","24576","4096"],["327680","1048576","1179648","524288","65536"],["6291456","26214400","41943040","31457280","10485760","1048576"]],"row_sums_preview":["2048","81920","3145728","117440512","4294967296","154618822656","5497558138880","193514046488576"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a93042cd70c4761e","steps":["basis A013611","D+1"],"edges":["e-dab838a9194ad2c3","e-5041fe3f32906394"],"text":"A038330 \u2192 basis A013611 \u2192 D+1"},"headline":"A038330 \u2192 basis A013611 \u2192 D+1","chain":null}},{"id":"l-2b0ed8f1aba57f3e","kind":"fingerprint_related","node_a_id":"n-1e3d729fffbceff3","node_b_id":"n-8e8138bcc83db1eb","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-8e8138bcc83db1eb","other":{"id":"n-8e8138bcc83db1eb","name":"one_plus_derivative of Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*11^j.","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["22","11"],["363","484","121"]],"card_rows":[["1"],["22","11"],["363","484","121"],["5324","11979","7986","1331"]],"row_sums_preview":["1","33","968","26620","702768","18037712","453519616","11224610496"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8064e72cf80883a7","steps":["D+1"],"edges":["e-2c0bf804848dbc22"],"text":"A038325 \u2192 D+1"},"headline":"A038325 \u2192 D+1","chain":null}},{"id":"l-2efd2692e701dbc2","kind":"fingerprint_related","node_a_id":"n-777049cbeb01df27","node_b_id":"n-1e3d729fffbceff3","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-777049cbeb01df27","other":{"id":"n-777049cbeb01df27","name":"theta_plus_one of Triangle whose (i,j)-th entry is binomial(i,j)*2^i.","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","4"],["4","16","12"]],"card_rows":[["1"],["2","4"],["4","16","12"],["8","48","72","32"]],"row_sums_preview":["1","6","32","160","768","3584","16384","73728"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d28ad9841fc73317","steps":["\u03b8+1"],"edges":["e-0211c2ba010af36a"],"text":"A038208 \u2192 \u03b8+1"},"headline":"A038208 \u2192 \u03b8+1","chain":null}},{"id":"l-3473055d6ae4e030","kind":"fingerprint_related","node_a_id":"n-1e3d729fffbceff3","node_b_id":"n-c73a1014825b0ccc","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-c73a1014825b0ccc","other":{"id":"n-c73a1014825b0ccc","name":"theta_plus_one of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["144","576","432"],["1728","10368","15552","6912"],["20736","165888","373248","331776","103680"]],"card_rows":[["144","576","432"],["1728","10368","15552","6912"],["20736","165888","373248","331776","103680"],["248832","2488320","7464960","9953280","6220800","1492992"]],"row_sums_preview":["1152","34560","995328","27869184","764411904","20639121408","550376570880","14529941471232"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c1532953a6736702","steps":["\u03b8+1"],"edges":["e-8755bd3e47da1469"],"text":"A038338 \u2192 \u03b8+1"},"headline":"A038338 \u2192 \u03b8+1","chain":null}},{"id":"l-360e72a23e47a265","kind":"fingerprint_related","node_a_id":"n-922a52ffa8f2e057","node_b_id":"n-1e3d729fffbceff3","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-922a52ffa8f2e057","other":{"id":"n-922a52ffa8f2e057","name":"one_plus_derivative of Exponential Riordan array [x*exp(x),x].","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],["4","2"]],"card_rows":[["1"],["4","2"],["9","12","3"],["16","36","24","4"]],"row_sums_preview":["0","1","6","24","80","240","672","1792"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f24dc5f77f658331","steps":["D"],"edges":["e-2695bc03a925fb68"],"text":"A093375 \u2192 D"},"headline":"A093375 \u2192 D","chain":null}},{"id":"l-384de39b8afb4496","kind":"fingerprint_related","node_a_id":"n-1e3d729fffbceff3","node_b_id":"n-52b3f64dac2ca596","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-52b3f64dac2ca596","other":{"id":"n-52b3f64dac2ca596","name":"one_plus_derivative of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(n).","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","1"],["9","12","3"]],"card_rows":[["1"],["2","1"],["9","12","3"],["60","135","90","15"]],"row_sums_preview":["1","3","24","300","5040","105840","2661120","77837760"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c7809537965f400d","steps":["D+1"],"edges":["e-f9eb8f23471ae7fc"],"text":"A085881 \u2192 D+1"},"headline":"A085881 \u2192 D+1","chain":null}},{"id":"l-3b2c6e4c51c1b594","kind":"fingerprint_related","node_a_id":"n-3096119ab2c8ef2b","node_b_id":"n-1e3d729fffbceff3","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-3096119ab2c8ef2b","other":{"id":"n-3096119ab2c8ef2b","name":"theta_plus_one of A Jacobsthal-Pascal triangle.","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","2"],["3","12","9"]],"card_rows":[["1"],["1","2"],["3","12","9"],["5","30","45","20"]],"row_sums_preview":["1","3","24","100","528","2352","11008","48960"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-9f2158ef8512cadd","steps":["\u03b8+1"],"edges":["e-fb0050d910f72695"],"text":"A124860 \u2192 \u03b8+1"},"headline":"A124860 \u2192 \u03b8+1","chain":null}},{"id":"l-4236341388855c01","kind":"fingerprint_related","node_a_id":"n-609aa611c94bbe52","node_b_id":"n-1e3d729fffbceff3","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-609aa611c94bbe52","other":{"id":"n-609aa611c94bbe52","name":"one_plus_derivative of Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["12","6"],["63","84","21"]],"card_rows":[["1"],["12","6"],["63","84","21"],["224","504","336","56"]],"row_sums_preview":["1","18","168","1120","6048","28224","118272","456192"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-97228a88c81561dc","steps":["D+1"],"edges":["e-0c1fb63d7f7db171"],"text":"A178822 \u2192 D+1"},"headline":"A178822 \u2192 D+1","chain":null}},{"id":"l-441d0739f560359e","kind":"fingerprint_related","node_a_id":"n-f24dc5f77f658331","node_b_id":"n-1e3d729fffbceff3","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-f24dc5f77f658331","other":{"id":"n-f24dc5f77f658331","name":"Array T(m,n) read by ascending antidiagonals: T(m,n) = m*binomial(n+m-2, n-1) for m, n >= 1.","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","1"],["3","4","1"]],"card_rows":[["1"],["2","1"],["3","4","1"],["4","9","6","1"]],"row_sums_preview":["1","3","8","20","48","112","256","576"],"regions":["(-inf, 0]","(-inf, 0)"],"region_exact":{},"oeis_ids":["A093375"],"derivation":{"root":"n-f24dc5f77f658331","steps":[],"edges":[],"text":"A093375"},"headline":"A093375: Array T(m,n) read by ascending antidiagonals: T(m,n) = m*binomial(n+m-2, n-1) for m, n >= 1.","chain":null}},{"id":"l-44e081d937f30385","kind":"fingerprint_related","node_a_id":"n-ae892ec0e2e55006","node_b_id":"n-1e3d729fffbceff3","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-ae892ec0e2e55006","other":{"id":"n-ae892ec0e2e55006","name":"one_plus_derivative of derivative of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["2"],["8","4"]],"card_rows":[["2"],["8","4"],["18","24","6"],["32","72","48","8"]],"row_sums_preview":["0","2","12","48","160","480","1344","3584"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-089a9b4f824a1c20","steps":["D","D+1"],"edges":["e-509720341f277f4c","e-d41cbe4d18614293"],"text":"A028326 \u2192 D \u2192 D+1"},"headline":"A028326 \u2192 D \u2192 D+1","chain":null}},{"id":"l-48401960d54e6add","kind":"fingerprint_related","node_a_id":"n-124bc02fba57fd6b","node_b_id":"n-1e3d729fffbceff3","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-124bc02fba57fd6b","other":{"id":"n-124bc02fba57fd6b","name":"theta_plus_one of derivative of Exponential Riordan array [x*exp(x),x].","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],[],["2"]],"card_rows":[["2"],["6","12"],["12","48","36"],["20","120","180","80"]],"row_sums_preview":["0","0","2","18","96","400","1440","4704"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7b5887f058255aa1","steps":["D","\u03b8+1"],"edges":["e-0d0bdaee7e5f71e7","e-92091e49aded2a3c"],"text":"A003506 \u2192 D \u2192 \u03b8+1"},"headline":"A003506 \u2192 D \u2192 \u03b8+1","chain":null}},{"id":"l-4c6131f1ba9153e3","kind":"fingerprint_related","node_a_id":"n-1e3d729fffbceff3","node_b_id":"n-bf0d178eef6b2fcb","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-bf0d178eef6b2fcb","other":{"id":"n-bf0d178eef6b2fcb","name":"theta_plus_one of Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","2"],["2","8","6"]],"card_rows":[["1"],["1","2"],["2","8","6"],["6","36","54","24"]],"row_sums_preview":["1","3","16","120","1152","13440","184320","2903040"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cb5d49e71e46f2ed","steps":["\u03b8+1"],"edges":["e-45ee6c7703cfa5e7"],"text":"A196347 \u2192 \u03b8+1"},"headline":"A196347 \u2192 \u03b8+1","chain":null}},{"id":"l-4cf22fccfee0386a","kind":"fingerprint_related","node_a_id":"n-1e3d729fffbceff3","node_b_id":"n-3645655eae90491b","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-3645655eae90491b","other":{"id":"n-3645655eae90491b","name":"one_plus_derivative of basis_map_A013609 of Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["24","12"],["252","336","84"]],"card_rows":[["1"],["24","12"],["252","336","84"],["1792","4032","2688","448"]],"row_sums_preview":["1","36","672","8960","96768","903168","7569408","58392576"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-97228a88c81561dc","steps":["basis A013609","D+1"],"edges":["e-5270021ea4ce6d61","e-c3b24e53a82407a6"],"text":"A178822 \u2192 basis A013609 \u2192 D+1"},"headline":"A178822 \u2192 basis A013609 \u2192 D+1","chain":null}},{"id":"l-4ed1f3629befaffb","kind":"fingerprint_related","node_a_id":"n-1e3d729fffbceff3","node_b_id":"n-a415cf3e2600f94b","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-a415cf3e2600f94b","other":{"id":"n-a415cf3e2600f94b","name":"theta_plus_one of Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*11^j.","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["11","22"],["121","484","363"]],"card_rows":[["1"],["11","22"],["121","484","363"],["1331","7986","11979","5324"]],"row_sums_preview":["1","33","968","26620","702768","18037712","453519616","11224610496"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8064e72cf80883a7","steps":["\u03b8+1"],"edges":["e-aec19f6e96f665f4"],"text":"A038325 \u2192 \u03b8+1"},"headline":"A038325 \u2192 \u03b8+1","chain":null}},{"id":"l-52bcf490731e0d1e","kind":"fingerprint_related","node_a_id":"n-a00a29c17e5648c7","node_b_id":"n-1e3d729fffbceff3","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-a00a29c17e5648c7","other":{"id":"n-a00a29c17e5648c7","name":"one_plus_derivative of Triangle read by rows, for 0 <= k <= n: T(n, k) = binomial(n, k) * binomial(binomial(n + 3, 2), 2).","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["3"],["30","15"],["135","180","45"]],"card_rows":[["3"],["30","15"],["135","180","45"],["420","945","630","105"]],"row_sums_preview":["3","45","360","2100","10080","42336","161280","570240"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-dd4aa31f3d11d152","steps":["D+1"],"edges":["e-711d9c55c7b6eabe"],"text":"A344565 \u2192 D+1"},"headline":"A344565 \u2192 D+1","chain":null}},{"id":"l-52c02260edcedf8e","kind":"fingerprint_related","node_a_id":"n-1e3d729fffbceff3","node_b_id":"n-280fc6ef9084bfbe","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-280fc6ef9084bfbe","other":{"id":"n-280fc6ef9084bfbe","name":"one_plus_derivative of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*3^j.","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["12","6"],["108","144","36"]],"card_rows":[["1"],["12","6"],["108","144","36"],["864","1944","1296","216"]],"row_sums_preview":["1","18","288","4320","62208","870912","11943936","161243136"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8d88ef9de74b305f","steps":["basis A013609","D+1"],"edges":["e-86b1b19edf92ec9a","e-4fdb3cd75016c5af"],"text":"A038221 \u2192 basis A013609 \u2192 D+1"},"headline":"A038221 \u2192 basis A013609 \u2192 D+1","chain":null}},{"id":"l-52cd4822a7bbec10","kind":"fingerprint_related","node_a_id":"n-1e3d729fffbceff3","node_b_id":"n-d34d6a3cf20a6764","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-d34d6a3cf20a6764","other":{"id":"n-d34d6a3cf20a6764","name":"one_plus_derivative of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["432","576","144"],["6912","15552","10368","1728"],["103680","331776","373248","165888","20736"]],"card_rows":[["432","576","144"],["6912","15552","10368","1728"],["103680","331776","373248","165888","20736"],["1492992","6220800","9953280","7464960","2488320","248832"]],"row_sums_preview":["1152","34560","995328","27869184","764411904","20639121408","550376570880","14529941471232"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c1532953a6736702","steps":["D+1"],"edges":["e-36dcb952c4a67e48"],"text":"A038338 \u2192 D+1"},"headline":"A038338 \u2192 D+1","chain":null}},{"id":"l-5a23a62880a0a220","kind":"fingerprint_related","node_a_id":"n-1e3d729fffbceff3","node_b_id":"n-8b97a6525df27660","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-8b97a6525df27660","other":{"id":"n-8b97a6525df27660","name":"theta_plus_one of basis_map_A013609 of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(n).","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","4"],["12","48","36"]],"card_rows":[["1"],["2","4"],["12","48","36"],["120","720","1080","480"]],"row_sums_preview":["1","6","96","2400","80640","3386880","170311680","9963233280"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c7809537965f400d","steps":["basis A013609","\u03b8+1"],"edges":["e-0f3734a094503340","e-84d7a761adafebf0"],"text":"A085881 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A085881 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}},{"id":"l-5c7ad8406fd88aa5","kind":"fingerprint_related","node_a_id":"n-1e3d729fffbceff3","node_b_id":"n-1bdcd86a0213c685","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-1bdcd86a0213c685","other":{"id":"n-1bdcd86a0213c685","name":"theta_plus_one of basis_map_A013609 of Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","16"],["40","160","120"]],"card_rows":[["1"],["8","16"],["40","160","120"],["160","960","1440","640"]],"row_sums_preview":["1","24","320","3200","26880","200704","1376256","8847360"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a820c1859f68e71c","steps":["basis A013609","\u03b8+1"],"edges":["e-9fe823de1e8617e8","e-482f9c74ebff9bd6"],"text":"A178820 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A178820 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}},{"id":"l-5de363814f34eb49","kind":"fingerprint_related","node_a_id":"n-2685bfce477ea338","node_b_id":"n-1e3d729fffbceff3","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-2685bfce477ea338","other":{"id":"n-2685bfce477ea338","name":"one_plus_derivative of basis-map coefficients of Triangle read by rows: row n consists of n-1 zeros followed by Fibonacci(n).","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","1"],["6","8","2"]],"card_rows":[["1"],["2","1"],["6","8","2"],["12","27","18","3"]],"row_sums_preview":["1","3","16","60","240","896","3328","12096"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ed81f100961f7473","steps":["D+1"],"edges":["e-fe095a79c59ce630"],"text":"basis-map coefficients of A127647 \u2192 D+1"},"headline":"basis-map coefficients of A127647 \u2192 D+1","chain":null}},{"id":"l-60dfc51cca427cf3","kind":"fingerprint_related","node_a_id":"n-139c09f162fa0372","node_b_id":"n-1e3d729fffbceff3","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-139c09f162fa0372","other":{"id":"n-139c09f162fa0372","name":"theta_plus_one of basis-map coefficients of Triangle read by rows: row n consists of n-1 zeros followed by Fibonacci(n).","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","2"],["2","8","6"]],"card_rows":[["1"],["1","2"],["2","8","6"],["3","18","27","12"]],"row_sums_preview":["1","3","16","60","240","896","3328","12096"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ed81f100961f7473","steps":["\u03b8+1"],"edges":["e-43245b5c332852b3"],"text":"basis-map coefficients of A127647 \u2192 \u03b8+1"},"headline":"basis-map coefficients of A127647 \u2192 \u03b8+1","chain":null}},{"id":"l-62d509708b4b0fdc","kind":"fingerprint_related","node_a_id":"n-1e3d729fffbceff3","node_b_id":"n-3303313970cb18fb","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-3303313970cb18fb","other":{"id":"n-3303313970cb18fb","name":"theta_plus_one of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*2^i.","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","8"],["16","64","48"]],"card_rows":[["1"],["4","8"],["16","64","48"],["64","384","576","256"]],"row_sums_preview":["1","12","128","1280","12288","114688","1048576","9437184"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8d887bfc40d036b9","steps":["basis A013611","\u03b8+1"],"edges":["e-b3c62654af8090af","e-0b414474b715b010"],"text":"A027465 \u2192 basis A013611 \u2192 \u03b8+1"},"headline":"A027465 \u2192 basis A013611 \u2192 \u03b8+1","chain":null}},{"id":"l-637a098e7d46e117","kind":"fingerprint_related","node_a_id":"n-1abb04c9738b703c","node_b_id":"n-1e3d729fffbceff3","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-1abb04c9738b703c","other":{"id":"n-1abb04c9738b703c","name":"theta_plus_one of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).","first_index":1,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","2"],["2","8","6"]],"card_rows":[["1"],["1","2"],["2","8","6"],["5","30","45","20"]],"row_sums_preview":["1","3","16","100","672","4704","33792","247104"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a98fae1bf9eeb913","steps":["\u03b8+1"],"edges":["e-1e4b7dd7009119da"],"text":"A085880 \u2192 \u03b8+1"},"headline":"A085880 \u2192 \u03b8+1","chain":null}},{"id":"l-6673af47fcd7e3f5","kind":"fingerprint_related","node_a_id":"n-d5120fbfd4c4c97a","node_b_id":"n-1e3d729fffbceff3","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-d5120fbfd4c4c97a","other":{"id":"n-d5120fbfd4c4c97a","name":"theta_plus_one of Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","4"],["6","24","18"]],"card_rows":[["1"],["2","4"],["6","24","18"],["20","120","180","80"]],"row_sums_preview":["1","6","48","400","3360","28224","236544","1976832"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-dd77bf712b1df996","steps":["\u03b8+1"],"edges":["e-72737e709303e989"],"text":"A356546 \u2192 \u03b8+1"},"headline":"A356546 \u2192 \u03b8+1","chain":null}},{"id":"l-66c1c28829c13ac2","kind":"fingerprint_related","node_a_id":"n-da71e01126b2b035","node_b_id":"n-1e3d729fffbceff3","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-da71e01126b2b035","other":{"id":"n-da71e01126b2b035","name":"theta_plus_one of Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["5","10"],["15","60","45"]],"card_rows":[["1"],["5","10"],["15","60","45"],["35","210","315","140"]],"row_sums_preview":["1","15","120","700","3360","14112","53760","190080"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-79986ab5023c2764","steps":["\u03b8+1"],"edges":["e-597039afcb309e3c"],"text":"A178821 \u2192 \u03b8+1"},"headline":"A178821 \u2192 \u03b8+1","chain":null}},{"id":"l-6b3294a00fd915bb","kind":"fingerprint_related","node_a_id":"n-6e023f035bca0041","node_b_id":"n-1e3d729fffbceff3","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-6e023f035bca0041","other":{"id":"n-6e023f035bca0041","name":"one_plus_derivative of Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","4"],["30","40","10"]],"card_rows":[["1"],["8","4"],["30","40","10"],["80","180","120","20"]],"row_sums_preview":["1","12","80","400","1680","6272","21504","69120"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a820c1859f68e71c","steps":["D+1"],"edges":["e-70e8f707f44918cf"],"text":"A178820 \u2192 D+1"},"headline":"A178820 \u2192 D+1","chain":null}},{"id":"l-6e32a30a3a05bea8","kind":"fingerprint_related","node_a_id":"n-1e3d729fffbceff3","node_b_id":"n-ac9d29ad31c588cb","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-ac9d29ad31c588cb","other":{"id":"n-ac9d29ad31c588cb","name":"one_plus_derivative of basis_map_A013609 of Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*11^j.","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["44","22"],["1452","1936","484"]],"card_rows":[["1"],["44","22"],["1452","1936","484"],["42592","95832","63888","10648"]],"row_sums_preview":["1","66","3872","212960","11244288","577206784","29025255424","1436750143488"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8064e72cf80883a7","steps":["basis A013609","D+1"],"edges":["e-290484f08292dc0a","e-c10af4e5f3828be8"],"text":"A038325 \u2192 basis A013609 \u2192 D+1"},"headline":"A038325 \u2192 basis A013609 \u2192 D+1","chain":null}},{"id":"l-6e87a3316b379d71","kind":"fingerprint_related","node_a_id":"n-1e3d729fffbceff3","node_b_id":"n-7eb70857c2b50e89","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-7eb70857c2b50e89","other":{"id":"n-7eb70857c2b50e89","name":"theta_plus_one of basis_map_A013609 of Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","4"],["8","32","24"]],"card_rows":[["1"],["2","4"],["8","32","24"],["48","288","432","192"]],"row_sums_preview":["1","6","64","960","18432","430080","11796480","371589120"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cb5d49e71e46f2ed","steps":["basis A013609","\u03b8+1"],"edges":["e-6d279c69a37931ca","e-e7c23bde1a3120fb"],"text":"A196347 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A196347 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}},{"id":"l-7131630296a9c40a","kind":"fingerprint_related","node_a_id":"n-7a5b2f8a6a89e281","node_b_id":"n-1e3d729fffbceff3","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-7a5b2f8a6a89e281","other":{"id":"n-7a5b2f8a6a89e281","name":"theta_plus_one of T(n, k) = binomial(n, k)*hypergeom([(1 - n)/2, -n/2], [1], 4).","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","2"],["3","12","9"]],"card_rows":[["1"],["1","2"],["3","12","9"],["7","42","63","28"]],"row_sums_preview":["1","3","24","140","912","5712","36096","226368"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-63b9a6ee72661305","steps":["\u03b8+1"],"edges":["e-4e295c11d518255b"],"text":"A376827 \u2192 \u03b8+1"},"headline":"A376827 \u2192 \u03b8+1","chain":null}},{"id":"l-71d0a4099fdae1ce","kind":"fingerprint_related","node_a_id":"n-8c3fe1c3189f5b0e","node_b_id":"n-1e3d729fffbceff3","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-8c3fe1c3189f5b0e","other":{"id":"n-8c3fe1c3189f5b0e","name":"one_plus_derivative of Triangle read by rows: T(n,k) = 2^floor(n/2)*binomial(n-1,k-1).","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","2"],["6","8","2"]],"card_rows":[["1"],["4","2"],["6","8","2"],["16","36","24","4"]],"row_sums_preview":["1","6","16","80","192","896","2048","9216"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a624e5aba8e1c39e","steps":["D+1"],"edges":["e-40cfcf3b0ff2ebc3"],"text":"A135838 \u2192 D+1"},"headline":"A135838 \u2192 D+1","chain":null}},{"id":"l-739aa3f3a55ac2d5","kind":"fingerprint_related","node_a_id":"n-ac5c1e9a0fbadb48","node_b_id":"n-1e3d729fffbceff3","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-ac5c1e9a0fbadb48","other":{"id":"n-ac5c1e9a0fbadb48","name":"one_plus_derivative of basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-2","-1"],["3","4","1"]],"card_rows":[["1"],["-2","-1"],["3","4","1"],["-4","-9","-6","-1"]],"row_sums_preview":["1","-3","8","-20","48","-112","256","-576"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-74c1e94669ff21ff","steps":["D+1"],"edges":["e-3b50ee0d45bb8d84"],"text":"basis-map coefficients of A113214, shifted \u2192 D+1"},"headline":"basis-map coefficients of A113214, shifted \u2192 D+1","chain":null}},{"id":"l-77286b5fcaafd14d","kind":"fingerprint_related","node_a_id":"n-00f17f8389bd6548","node_b_id":"n-1e3d729fffbceff3","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-00f17f8389bd6548","other":{"id":"n-00f17f8389bd6548","name":"theta_plus_one of shift_x_plus_one of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","first_index":1,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","2"],["1","4","3"]],"card_rows":[["1"],["1","2"],["1","4","3"],["1","6","9","4"]],"row_sums_preview":["1","3","8","20","48","112","256","576"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1bf99ad40f056d33","steps":["basis A007318","\u03b8+1"],"edges":["e-279353e75dd63f59","e-6b513dac3e56c817"],"text":"basis-map coefficients of A026729 \u2192 basis A007318 \u2192 \u03b8+1"},"headline":"basis-map coefficients of A026729 \u2192 basis A007318 \u2192 \u03b8+1","chain":null}},{"id":"l-811f91cbe7995df5","kind":"fingerprint_related","node_a_id":"n-925258ea4d4afe84","node_b_id":"n-1e3d729fffbceff3","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-925258ea4d4afe84","other":{"id":"n-925258ea4d4afe84","name":"theta_plus_one of Triangle read by rows: T(n,k) = 2^floor(n/2)*binomial(n-1,k-1).","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","4"],["2","8","6"]],"card_rows":[["1"],["2","4"],["2","8","6"],["4","24","36","16"]],"row_sums_preview":["1","6","16","80","192","896","2048","9216"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a624e5aba8e1c39e","steps":["\u03b8+1"],"edges":["e-999c61cd00d73b61"],"text":"A135838 \u2192 \u03b8+1"},"headline":"A135838 \u2192 \u03b8+1","chain":null}},{"id":"l-8289d56020506122","kind":"fingerprint_related","node_a_id":"n-1e3d729fffbceff3","node_b_id":"n-3d8375aa67491629","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-3d8375aa67491629","other":{"id":"n-3d8375aa67491629","name":"one_plus_derivative of basis_map_A013609 of Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","2"],["24","32","8"]],"card_rows":[["1"],["4","2"],["24","32","8"],["192","432","288","48"]],"row_sums_preview":["1","6","64","960","18432","430080","11796480","371589120"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cb5d49e71e46f2ed","steps":["basis A013609","D+1"],"edges":["e-6d279c69a37931ca","e-1546b6157f5dd90b"],"text":"A196347 \u2192 basis A013609 \u2192 D+1"},"headline":"A196347 \u2192 basis A013609 \u2192 D+1","chain":null}},{"id":"l-83213b19e0725465","kind":"fingerprint_related","node_a_id":"n-1e3d729fffbceff3","node_b_id":"n-396b70e4bb089dfb","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-396b70e4bb089dfb","other":{"id":"n-396b70e4bb089dfb","name":"theta_plus_one of basis_map_A013609 of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).","first_index":1,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","4"],["8","32","24"]],"card_rows":[["1"],["2","4"],["8","32","24"],["40","240","360","160"]],"row_sums_preview":["1","6","64","800","10752","150528","2162688","31629312"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a98fae1bf9eeb913","steps":["basis A013609","\u03b8+1"],"edges":["e-ccc293aee69469d4","e-e935b98bfcf5f8b8"],"text":"A085880 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A085880 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}},{"id":"l-846c49246f908ed5","kind":"fingerprint_related","node_a_id":"n-1e3d729fffbceff3","node_b_id":"n-cd5e4a05afe9599d","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-cd5e4a05afe9599d","other":{"id":"n-cd5e4a05afe9599d","name":"theta_plus_one of basis_map_A013609 of Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","8"],["24","96","72"]],"card_rows":[["1"],["4","8"],["24","96","72"],["160","960","1440","640"]],"row_sums_preview":["1","12","192","3200","53760","903168","15138816","253034496"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-dd77bf712b1df996","steps":["basis A013609","\u03b8+1"],"edges":["e-2e1c22b656cc26bb","e-5795d07f40374c04"],"text":"A356546 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A356546 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}},{"id":"l-86b81073ab9a9077","kind":"fingerprint_related","node_a_id":"n-751f2fc33fb80fef","node_b_id":"n-1e3d729fffbceff3","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-751f2fc33fb80fef","other":{"id":"n-751f2fc33fb80fef","name":"theta_plus_one of Triangle read by rows: T(n,k) = ((n+1)(n+2)/2) * binomial(n,k) (0 <= k <= n).","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","6"],["6","24","18"]],"card_rows":[["1"],["3","6"],["6","24","18"],["10","60","90","40"]],"row_sums_preview":["1","9","48","200","720","2352","7168","20736"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7f1b09730105c8f0","steps":["\u03b8+1"],"edges":["e-463d07cf5504ee11"],"text":"A094305 \u2192 \u03b8+1"},"headline":"A094305 \u2192 \u03b8+1","chain":null}},{"id":"l-8d1c5d8271f7cdcb","kind":"fingerprint_related","node_a_id":"n-441d800dafd42935","node_b_id":"n-1e3d729fffbceff3","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-441d800dafd42935","other":{"id":"n-441d800dafd42935","name":"one_plus_derivative of A127733 * A007318 as infinite lower triangular matrices.","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","4"],["27","36","9"]],"card_rows":[["1"],["8","4"],["27","36","9"],["64","144","96","16"]],"row_sums_preview":["1","12","72","320","1200","4032","12544","36864"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-375ba152c4aebf61","steps":["D+1"],"edges":["e-7e3f1cd4b08a4540"],"text":"A135065 \u2192 D+1"},"headline":"A135065 \u2192 D+1","chain":null}},{"id":"l-8d86d65f7944e463","kind":"fingerprint_related","node_a_id":"n-1cd5d4ff73ee9007","node_b_id":"n-1e3d729fffbceff3","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-1cd5d4ff73ee9007","other":{"id":"n-1cd5d4ff73ee9007","name":"one_plus_derivative of derivative of Exponential Riordan array [x*exp(x),x].","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],[],["2"]],"card_rows":[["2"],["12","6"],["36","48","12"],["80","180","120","20"]],"row_sums_preview":["0","0","2","18","96","400","1440","4704"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f24dc5f77f658331","steps":["D","D"],"edges":["e-2695bc03a925fb68","e-0e4c420a090aad2c"],"text":"A093375 \u2192 D \u2192 D"},"headline":"A093375 \u2192 D \u2192 D","chain":null}},{"id":"l-9048e8674716a588","kind":"fingerprint_related","node_a_id":"n-e78214c646c97ace","node_b_id":"n-1e3d729fffbceff3","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-e78214c646c97ace","other":{"id":"n-e78214c646c97ace","name":"theta_plus_one of Triangular array read by rows: T(n,k) = binomial(n,2)*binomial(n,k), n>=0, 0<=k<=n.","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],[],["1","4","3"]],"card_rows":[["1","4","3"],["3","18","27","12"],["6","48","108","96","30"],["10","100","300","400","250","60"]],"row_sums_preview":["0","0","8","60","288","1120","3840","12096"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c8c19eba9d68881f","steps":["\u00d7(\u2212(1/2)n+(1/2)n\u00b2)","\u03b8+1"],"edges":["e-ccfad80d3bf960c1","e-21ef48566d5646a5"],"text":"A007318 \u2192 \u00d7(\u2212(1/2)n+(1/2)n\u00b2) \u2192 \u03b8+1"},"headline":"A007318 \u2192 \u00d7(\u2212(1/2)n+(1/2)n\u00b2) \u2192 \u03b8+1","chain":null}},{"id":"l-918762d474cf32c8","kind":"fingerprint_related","node_a_id":"n-1e3d729fffbceff3","node_b_id":"n-76c02e9a4178807b","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-76c02e9a4178807b","other":{"id":"n-76c02e9a4178807b","name":"one_plus_derivative of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*2^i.","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","4"],["48","64","16"]],"card_rows":[["1"],["8","4"],["48","64","16"],["256","576","384","64"]],"row_sums_preview":["1","12","128","1280","12288","114688","1048576","9437184"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8d887bfc40d036b9","steps":["basis A013611","D+1"],"edges":["e-b3c62654af8090af","e-cf29354f5b1fc249"],"text":"A027465 \u2192 basis A013611 \u2192 D+1"},"headline":"A027465 \u2192 basis A013611 \u2192 D+1","chain":null}},{"id":"l-97579a64bc4fc545","kind":"fingerprint_related","node_a_id":"n-e7b37aaa21296aa0","node_b_id":"n-1e3d729fffbceff3","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-e7b37aaa21296aa0","other":{"id":"n-e7b37aaa21296aa0","name":"one_plus_derivative of shift_x_plus_one of basis-map coefficients of A scaled Hermite triangle.","first_index":1,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["3","4","1"],["4","9","6","1"],["5","16","18","8","1"]],"card_rows":[["3","4","1"],["4","9","6","1"],["5","16","18","8","1"],["6","25","40","30","10","1"]],"row_sums_preview":["8","20","48","112","256","576","1280","2816"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5fe240122e98ffb3","steps":["x \u21a6 x+1","D+1"],"edges":["e-cb9b2e4b7b317e85","e-02626351e9b6be7e"],"text":"basis-map coefficients of A112227 \u2192 x \u21a6 x+1 \u2192 D+1"},"headline":"basis-map coefficients of A112227 \u2192 x \u21a6 x+1 \u2192 D+1","chain":null}},{"id":"l-99535d74bbc62504","kind":"fingerprint_related","node_a_id":"n-1e3d729fffbceff3","node_b_id":"n-25474d00a8e4fbc9","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-25474d00a8e4fbc9","other":{"id":"n-25474d00a8e4fbc9","name":"theta_plus_one of basis_map_A013609 of Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","8"],["24","96","72"]],"card_rows":[["1"],["4","8"],["24","96","72"],["192","1152","1728","768"]],"row_sums_preview":["1","12","192","3840","92160","2580480","82575360","2972712960"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-75a6475b3e851dc5","steps":["basis A013609","\u03b8+1"],"edges":["e-b7289e9f518eca68","e-6e74a790fd22a254"],"text":"A156992 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A156992 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}},{"id":"l-9a5581f5e43c6d6b","kind":"fingerprint_related","node_a_id":"n-1e3d729fffbceff3","node_b_id":"n-ad1d003821d5e8ad","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-ad1d003821d5e8ad","other":{"id":"n-ad1d003821d5e8ad","name":"one_plus_derivative of basis_map_A013609 of Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","4"],["72","96","24"]],"card_rows":[["1"],["8","4"],["72","96","24"],["640","1440","960","160"]],"row_sums_preview":["1","12","192","3200","53760","903168","15138816","253034496"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-dd77bf712b1df996","steps":["basis A013609","D+1"],"edges":["e-2e1c22b656cc26bb","e-cddde93c1e6a9021"],"text":"A356546 \u2192 basis A013609 \u2192 D+1"},"headline":"A356546 \u2192 basis A013609 \u2192 D+1","chain":null}},{"id":"l-9aaa5a8796c85b9b","kind":"fingerprint_related","node_a_id":"n-6e0156c26c94d278","node_b_id":"n-1e3d729fffbceff3","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-6e0156c26c94d278","other":{"id":"n-6e0156c26c94d278","name":"theta_plus_one of shift_x_plus_one of basis-map coefficients of A scaled Hermite triangle.","first_index":1,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1","4","3"],["1","6","9","4"],["1","8","18","16","5"]],"card_rows":[["1","4","3"],["1","6","9","4"],["1","8","18","16","5"],["1","10","30","40","25","6"]],"row_sums_preview":["8","20","48","112","256","576","1280","2816"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5fe240122e98ffb3","steps":["x \u21a6 x+1","\u03b8+1"],"edges":["e-cb9b2e4b7b317e85","e-039ab71b88c09809"],"text":"basis-map coefficients of A112227 \u2192 x \u21a6 x+1 \u2192 \u03b8+1"},"headline":"basis-map coefficients of A112227 \u2192 x \u21a6 x+1 \u2192 \u03b8+1","chain":null}},{"id":"l-9eef82d0408e8ba9","kind":"fingerprint_related","node_a_id":"n-fbea127aefb17f0e","node_b_id":"n-1e3d729fffbceff3","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-fbea127aefb17f0e","other":{"id":"n-fbea127aefb17f0e","name":"theta_plus_one of basis_map_A013609 of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["2"],["4","8"],["8","32","24"]],"card_rows":[["2"],["4","8"],["8","32","24"],["16","96","144","64"]],"row_sums_preview":["2","12","64","320","1536","7168","32768","147456"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-089a9b4f824a1c20","steps":["basis A013609","\u03b8+1"],"edges":["e-23f81812d1886d3d","e-391772cd5cd0b48c"],"text":"A028326 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A028326 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}},{"id":"l-aa9ba19e929847af","kind":"fingerprint_related","node_a_id":"n-c14c1cf7f058cd0c","node_b_id":"n-1e3d729fffbceff3","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-c14c1cf7f058cd0c","other":{"id":"n-c14c1cf7f058cd0c","name":"theta_plus_one of basis_map_A013609 of basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-2","-4"],["4","16","12"]],"card_rows":[["1"],["-2","-4"],["4","16","12"],["-8","-48","-72","-32"]],"row_sums_preview":["1","-6","32","-160","768","-3584","16384","-73728"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-74c1e94669ff21ff","steps":["basis A013609","\u03b8+1"],"edges":["e-b6cdc8ab0f2a09ef","e-c28206d41632be9d"],"text":"basis-map coefficients of A113214, shifted \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"basis-map coefficients of A113214, shifted \u2192 basis A013609 \u2192 \u03b8+1","chain":null}},{"id":"l-abc0743d4af164c9","kind":"fingerprint_related","node_a_id":"n-1e3d729fffbceff3","node_b_id":"n-5f6b8ba9679b6006","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-5f6b8ba9679b6006","other":{"id":"n-5f6b8ba9679b6006","name":"theta_plus_one of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(n).","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","2"],["3","12","9"]],"card_rows":[["1"],["1","2"],["3","12","9"],["15","90","135","60"]],"row_sums_preview":["1","3","24","300","5040","105840","2661120","77837760"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c7809537965f400d","steps":["\u03b8+1"],"edges":["e-7a1f614597cef918"],"text":"A085881 \u2192 \u03b8+1"},"headline":"A085881 \u2192 \u03b8+1","chain":null}},{"id":"l-af3863dabced9489","kind":"fingerprint_related","node_a_id":"n-1e3d729fffbceff3","node_b_id":"n-5b0a78e5ea1764c3","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-5b0a78e5ea1764c3","other":{"id":"n-5b0a78e5ea1764c3","name":"theta_plus_one of basis_map_A013610 of Triangle read by rows: T(n,k) is the number of ternary words of length n having k levels (n >= 1, 0 <= k <= n-1). A level is a pair of identical consecutive letters.","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["3"],["9","18"],["27","108","81"]],"card_rows":[["3"],["9","18"],["27","108","81"],["81","486","729","324"]],"row_sums_preview":["3","27","216","1620","11664","81648","559872","3779136"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-51b4f5e7901c2b50","steps":["basis A013610","\u03b8+1"],"edges":["e-3ea97f7a94318269","e-06ca081b0c026faa"],"text":"A120910 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A120910 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}},{"id":"l-b02ed6c77daa8fd8","kind":"fingerprint_related","node_a_id":"n-6d32db1a89f2c14d","node_b_id":"n-1e3d729fffbceff3","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-6d32db1a89f2c14d","other":{"id":"n-6d32db1a89f2c14d","name":"theta_plus_one of shift_x_plus_one of 3n + 1 preceded by n zeros.","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","8"],["7","28","21"]],"card_rows":[["1"],["4","8"],["7","28","21"],["10","60","90","40"]],"row_sums_preview":["1","12","56","200","624","1792","4864","12672"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ccc86c3dcf2ac685","steps":["x \u21a6 x+1","\u03b8+1"],"edges":["e-28824f426538156a","e-b53841b10c461393"],"text":"A131431 \u2192 x \u21a6 x+1 \u2192 \u03b8+1"},"headline":"A131431 \u2192 x \u21a6 x+1 \u2192 \u03b8+1","chain":null}},{"id":"l-b06ef3061778f592","kind":"fingerprint_related","node_a_id":"n-2903ddeaf334f701","node_b_id":"n-1e3d729fffbceff3","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-2903ddeaf334f701","other":{"id":"n-2903ddeaf334f701","name":"theta_plus_one of Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*3^j.","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","6"],["9","36","27"]],"card_rows":[["1"],["3","6"],["9","36","27"],["27","162","243","108"]],"row_sums_preview":["1","9","72","540","3888","27216","186624","1259712"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8d88ef9de74b305f","steps":["\u03b8+1"],"edges":["e-285018a6642014ce"],"text":"A038221 \u2192 \u03b8+1"},"headline":"A038221 \u2192 \u03b8+1","chain":null}},{"id":"l-b1c803cf7bd49a30","kind":"fingerprint_related","node_a_id":"n-1e3d729fffbceff3","node_b_id":"n-ef56b61d1c966a30","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-ef56b61d1c966a30","other":{"id":"n-ef56b61d1c966a30","name":"theta_plus_one of basis_map_A013609 of Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["10","20"],["60","240","180"]],"card_rows":[["1"],["10","20"],["60","240","180"],["280","1680","2520","1120"]],"row_sums_preview":["1","30","480","5600","53760","451584","3440640","24330240"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-79986ab5023c2764","steps":["basis A013609","\u03b8+1"],"edges":["e-7c97406b114f1661","e-4dc0ad1ce9cc5ff6"],"text":"A178821 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A178821 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}},{"id":"l-b1e4d633ffda0980","kind":"fingerprint_related","node_a_id":"n-89ef366afa3c7a52","node_b_id":"n-1e3d729fffbceff3","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-89ef366afa3c7a52","other":{"id":"n-89ef366afa3c7a52","name":"theta_plus_one of Triangle read by rows: T(n,k) = (2^(n+1)-1)*binomial(n,k).","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","6"],["7","28","21"]],"card_rows":[["1"],["3","6"],["7","28","21"],["15","90","135","60"]],"row_sums_preview":["1","9","56","300","1488","7056","32512","146880"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8d96419786c3ee2a","steps":["\u03b8+1"],"edges":["e-6598a138023fb15b"],"text":"A134346 \u2192 \u03b8+1"},"headline":"A134346 \u2192 \u03b8+1","chain":null}},{"id":"l-b220be55612ae4ce","kind":"fingerprint_related","node_a_id":"n-37470e6a0ac8ffe2","node_b_id":"n-1e3d729fffbceff3","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-37470e6a0ac8ffe2","other":{"id":"n-37470e6a0ac8ffe2","name":"one_plus_derivative of shift_x_plus_one of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals. shifted by x -> -x + 0","first_index":1,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-2","-1"],["3","4","1"]],"card_rows":[["1"],["-2","-1"],["3","4","1"],["-4","-9","-6","-1"]],"row_sums_preview":["1","-3","8","-20","48","-112","256","-576"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3cc7875cf81a5a84","steps":["basis A007318","D+1"],"edges":["e-4c344817bd191837","e-e1b43047bbb132b8"],"text":"basis-map coefficients of A026729, shifted \u2192 basis A007318 \u2192 D+1"},"headline":"basis-map coefficients of A026729, shifted \u2192 basis A007318 \u2192 D+1","chain":null}},{"id":"l-b470db722cff3edd","kind":"fingerprint_related","node_a_id":"n-1e3d729fffbceff3","node_b_id":"n-7deea9a3b78b30cc","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-7deea9a3b78b30cc","other":{"id":"n-7deea9a3b78b30cc","name":"one_plus_derivative of basis_map_A013609 of Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["20","10"],["180","240","60"]],"card_rows":[["1"],["20","10"],["180","240","60"],["1120","2520","1680","280"]],"row_sums_preview":["1","30","480","5600","53760","451584","3440640","24330240"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-79986ab5023c2764","steps":["basis A013609","D+1"],"edges":["e-7c97406b114f1661","e-2bf7c06f4925e820"],"text":"A178821 \u2192 basis A013609 \u2192 D+1"},"headline":"A178821 \u2192 basis A013609 \u2192 D+1","chain":null}},{"id":"l-b56498adb357d333","kind":"fingerprint_related","node_a_id":"n-af70954370ea4182","node_b_id":"n-1e3d729fffbceff3","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-af70954370ea4182","other":{"id":"n-af70954370ea4182","name":"theta_plus_one of derivative of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["2"],["4","8"]],"card_rows":[["2"],["4","8"],["6","24","18"],["8","48","72","32"]],"row_sums_preview":["0","2","12","48","160","480","1344","3584"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-089a9b4f824a1c20","steps":["D","\u03b8+1"],"edges":["e-509720341f277f4c","e-4e4bd395a52d1454"],"text":"A028326 \u2192 D \u2192 \u03b8+1"},"headline":"A028326 \u2192 D \u2192 \u03b8+1","chain":null}},{"id":"l-b62d6c919fc24d00","kind":"fingerprint_related","node_a_id":"n-1e3d729fffbceff3","node_b_id":"n-9c4e830626e0fc99","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-9c4e830626e0fc99","other":{"id":"n-9c4e830626e0fc99","name":"theta_plus_one of basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["576","2304","1728"],["13824","82944","124416","55296"],["331776","2654208","5971968","5308416","1658880"]],"card_rows":[["576","2304","1728"],["13824","82944","124416","55296"],["331776","2654208","5971968","5308416","1658880"],["7962624","79626240","238878720","318504960","199065600","47775744"]],"row_sums_preview":["4608","276480","15925248","891813888","48922361856","2641807540224","140896402145280","7439330033270784"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c1532953a6736702","steps":["basis A013609","\u03b8+1"],"edges":["e-9ec880266e74c89d","e-3e121779d9d49914"],"text":"A038338 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A038338 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}},{"id":"l-b7b1c96b41fd60f3","kind":"fingerprint_related","node_a_id":"n-1e3d729fffbceff3","node_b_id":"n-2359edb2d01f321b","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-2359edb2d01f321b","other":{"id":"n-2359edb2d01f321b","name":"theta_plus_one of basis_map_A013609 of Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*11^j.","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["22","44"],["484","1936","1452"]],"card_rows":[["1"],["22","44"],["484","1936","1452"],["10648","63888","95832","42592"]],"row_sums_preview":["1","66","3872","212960","11244288","577206784","29025255424","1436750143488"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8064e72cf80883a7","steps":["basis A013609","\u03b8+1"],"edges":["e-290484f08292dc0a","e-6e08674843176c52"],"text":"A038325 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A038325 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}},{"id":"l-bb6aed05e2791e09","kind":"fingerprint_related","node_a_id":"n-1e3d729fffbceff3","node_b_id":"n-fe55273aeaac1673","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-fe55273aeaac1673","other":{"id":"n-fe55273aeaac1673","name":"theta_plus_one of Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","4"],["6","24","18"]],"card_rows":[["1"],["2","4"],["6","24","18"],["24","144","216","96"]],"row_sums_preview":["1","6","48","480","5760","80640","1290240","23224320"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-75a6475b3e851dc5","steps":["\u03b8+1"],"edges":["e-49d9b27036c079ef"],"text":"A156992 \u2192 \u03b8+1"},"headline":"A156992 \u2192 \u03b8+1","chain":null}},{"id":"l-bcd81b94efa15b24","kind":"fingerprint_related","node_a_id":"n-1e3d729fffbceff3","node_b_id":"n-aca43f562a3db171","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-aca43f562a3db171","other":{"id":"n-aca43f562a3db171","name":"one_plus_derivative of basis_map_A013609 of Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["16","8"],["120","160","40"]],"card_rows":[["1"],["16","8"],["120","160","40"],["640","1440","960","160"]],"row_sums_preview":["1","24","320","3200","26880","200704","1376256","8847360"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a820c1859f68e71c","steps":["basis A013609","D+1"],"edges":["e-9fe823de1e8617e8","e-c2c3b20968f920ac"],"text":"A178820 \u2192 basis A013609 \u2192 D+1"},"headline":"A178820 \u2192 basis A013609 \u2192 D+1","chain":null}},{"id":"l-bdd3db544504e944","kind":"fingerprint_related","node_a_id":"n-568ba9e14bf4b258","node_b_id":"n-1e3d729fffbceff3","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-568ba9e14bf4b258","other":{"id":"n-568ba9e14bf4b258","name":"theta_plus_one of shift_x_plus_one of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals. shifted by x -> -x + 0","first_index":1,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-1","-2"],["1","4","3"]],"card_rows":[["1"],["-1","-2"],["1","4","3"],["-1","-6","-9","-4"]],"row_sums_preview":["1","-3","8","-20","48","-112","256","-576"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3cc7875cf81a5a84","steps":["basis A007318","\u03b8+1"],"edges":["e-4c344817bd191837","e-3dd80bd93130705f"],"text":"basis-map coefficients of A026729, shifted \u2192 basis A007318 \u2192 \u03b8+1"},"headline":"basis-map coefficients of A026729, shifted \u2192 basis A007318 \u2192 \u03b8+1","chain":null}},{"id":"l-c2f6b9be46a80c5f","kind":"fingerprint_related","node_a_id":"n-3c5dcbc726f4972e","node_b_id":"n-1e3d729fffbceff3","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-3c5dcbc726f4972e","other":{"id":"n-3c5dcbc726f4972e","name":"one_plus_derivative of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).","first_index":1,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","1"],["6","8","2"]],"card_rows":[["1"],["2","1"],["6","8","2"],["20","45","30","5"]],"row_sums_preview":["1","3","16","100","672","4704","33792","247104"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a98fae1bf9eeb913","steps":["D+1"],"edges":["e-17a2449071a8dc49"],"text":"A085880 \u2192 D+1"},"headline":"A085880 \u2192 D+1","chain":null}},{"id":"l-c3dfda612f8ce6dd","kind":"fingerprint_related","node_a_id":"n-924179b433840212","node_b_id":"n-1e3d729fffbceff3","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-924179b433840212","other":{"id":"n-924179b433840212","name":"theta_plus_one of Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6","12"],["21","84","63"]],"card_rows":[["1"],["6","12"],["21","84","63"],["56","336","504","224"]],"row_sums_preview":["1","18","168","1120","6048","28224","118272","456192"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-97228a88c81561dc","steps":["\u03b8+1"],"edges":["e-6f1262911049e63e"],"text":"A178822 \u2192 \u03b8+1"},"headline":"A178822 \u2192 \u03b8+1","chain":null}},{"id":"l-c5c7d6b4f4cf2316","kind":"fingerprint_related","node_a_id":"n-d60a211de2f5ab9a","node_b_id":"n-1e3d729fffbceff3","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-d60a211de2f5ab9a","other":{"id":"n-d60a211de2f5ab9a","name":"one_plus_derivative of Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","2"],["18","24","6"]],"card_rows":[["1"],["4","2"],["18","24","6"],["80","180","120","20"]],"row_sums_preview":["1","6","48","400","3360","28224","236544","1976832"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-dd77bf712b1df996","steps":["D+1"],"edges":["e-782da903eb1c79dd"],"text":"A356546 \u2192 D+1"},"headline":"A356546 \u2192 D+1","chain":null}},{"id":"l-c60d5907f713f514","kind":"fingerprint_related","node_a_id":"n-a9e082eabd70d305","node_b_id":"n-1e3d729fffbceff3","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-a9e082eabd70d305","other":{"id":"n-a9e082eabd70d305","name":"theta_plus_one of A127733 * A007318 as infinite lower triangular matrices.","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","8"],["9","36","27"]],"card_rows":[["1"],["4","8"],["9","36","27"],["16","96","144","64"]],"row_sums_preview":["1","12","72","320","1200","4032","12544","36864"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-375ba152c4aebf61","steps":["\u03b8+1"],"edges":["e-703d2781a5e57ec4"],"text":"A135065 \u2192 \u03b8+1"},"headline":"A135065 \u2192 \u03b8+1","chain":null}},{"id":"l-cf574e0f626272f2","kind":"fingerprint_related","node_a_id":"n-b37b91e34380cdce","node_b_id":"n-1e3d729fffbceff3","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-b37b91e34380cdce","other":{"id":"n-b37b91e34380cdce","name":"one_plus_derivative of basis_map_A013609 of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["2"],["8","4"],["24","32","8"]],"card_rows":[["2"],["8","4"],["24","32","8"],["64","144","96","16"]],"row_sums_preview":["2","12","64","320","1536","7168","32768","147456"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-089a9b4f824a1c20","steps":["basis A013609","D+1"],"edges":["e-23f81812d1886d3d","e-ee54ab6262d25322"],"text":"A028326 \u2192 basis A013609 \u2192 D+1"},"headline":"A028326 \u2192 basis A013609 \u2192 D+1","chain":null}},{"id":"l-d27113f7caed233b","kind":"fingerprint_related","node_a_id":"n-1e3d729fffbceff3","node_b_id":"n-aec439afe31024dd","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-aec439afe31024dd","other":{"id":"n-aec439afe31024dd","name":"one_plus_derivative of basis_map_A013609 of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).","first_index":1,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","2"],["24","32","8"]],"card_rows":[["1"],["4","2"],["24","32","8"],["160","360","240","40"]],"row_sums_preview":["1","6","64","800","10752","150528","2162688","31629312"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a98fae1bf9eeb913","steps":["basis A013609","D+1"],"edges":["e-ccc293aee69469d4","e-305cc5fb3b95b0db"],"text":"A085880 \u2192 basis A013609 \u2192 D+1"},"headline":"A085880 \u2192 basis A013609 \u2192 D+1","chain":null}},{"id":"l-d53ea112191ccb61","kind":"fingerprint_related","node_a_id":"n-7faa5d30dd36de82","node_b_id":"n-1e3d729fffbceff3","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-7faa5d30dd36de82","other":{"id":"n-7faa5d30dd36de82","name":"theta_plus_one of derivative of Triangle whose (i,j)-th entry is binomial(i,j)*2^i.","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["2"],["8","16"]],"card_rows":[["2"],["8","16"],["24","96","72"],["64","384","576","256"]],"row_sums_preview":["0","2","24","192","1280","7680","43008","229376"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d28ad9841fc73317","steps":["D","\u03b8+1"],"edges":["e-a49f8ff31c907f66","e-0bc9faad932dd2e1"],"text":"A038208 \u2192 D \u2192 \u03b8+1"},"headline":"A038208 \u2192 D \u2192 \u03b8+1","chain":null}},{"id":"l-d6c8f03717f22833","kind":"fingerprint_related","node_a_id":"n-1e3d729fffbceff3","node_b_id":"n-22d41c489654ae69","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-22d41c489654ae69","other":{"id":"n-22d41c489654ae69","name":"one_plus_derivative of basis_map_A013609 of Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","4"],["72","96","24"]],"card_rows":[["1"],["8","4"],["72","96","24"],["768","1728","1152","192"]],"row_sums_preview":["1","12","192","3840","92160","2580480","82575360","2972712960"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-75a6475b3e851dc5","steps":["basis A013609","D+1"],"edges":["e-b7289e9f518eca68","e-cf86635f02de009a"],"text":"A156992 \u2192 basis A013609 \u2192 D+1"},"headline":"A156992 \u2192 basis A013609 \u2192 D+1","chain":null}},{"id":"l-d78e9264a65a1644","kind":"fingerprint_related","node_a_id":"n-0a3212c17664a261","node_b_id":"n-1e3d729fffbceff3","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-0a3212c17664a261","other":{"id":"n-0a3212c17664a261","name":"one_plus_derivative of T(n, k) = binomial(n, k)*hypergeom([(1 - n)/2, -n/2], [1], 4).","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","1"],["9","12","3"]],"card_rows":[["1"],["2","1"],["9","12","3"],["28","63","42","7"]],"row_sums_preview":["1","3","24","140","912","5712","36096","226368"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-63b9a6ee72661305","steps":["D+1"],"edges":["e-2f50b48903f23c2a"],"text":"A376827 \u2192 D+1"},"headline":"A376827 \u2192 D+1","chain":null}},{"id":"l-da086aa10a26d919","kind":"fingerprint_related","node_a_id":"n-8d978a50d5958248","node_b_id":"n-1e3d729fffbceff3","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-8d978a50d5958248","other":{"id":"n-8d978a50d5958248","name":"one_plus_derivative of derivative of basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[[],["-1"],["4","2"]],"card_rows":[["-1"],["4","2"],["-9","-12","-3"],["16","36","24","4"]],"row_sums_preview":["0","-1","6","-24","80","-240","672","-1792"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-74c1e94669ff21ff","steps":["D","D+1"],"edges":["e-3c99f80a26484c6d","e-bb63b6314f6baa7f"],"text":"basis-map coefficients of A113214, shifted \u2192 D \u2192 D+1"},"headline":"basis-map coefficients of A113214, shifted \u2192 D \u2192 D+1","chain":null}},{"id":"l-da816d11b7238c39","kind":"fingerprint_related","node_a_id":"n-84cf664a286120f4","node_b_id":"n-1e3d729fffbceff3","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-84cf664a286120f4","other":{"id":"n-84cf664a286120f4","name":"one_plus_derivative of derivative of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).","first_index":1,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],["8","4"]],"card_rows":[["1"],["8","4"],["45","60","15"],["224","504","336","56"]],"row_sums_preview":["0","1","12","120","1120","10080","88704","768768"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a98fae1bf9eeb913","steps":["D","D+1"],"edges":["e-592ee9fddf6f118c","e-1266d8844df0a516"],"text":"A085880 \u2192 D \u2192 D+1"},"headline":"A085880 \u2192 D \u2192 D+1","chain":null}},{"id":"l-dd35bd8387e4aca4","kind":"fingerprint_related","node_a_id":"n-9a960422aa31c822","node_b_id":"n-1e3d729fffbceff3","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-9a960422aa31c822","other":{"id":"n-9a960422aa31c822","name":"theta_plus_one of derivative of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).","first_index":1,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],["4","8"]],"card_rows":[["1"],["4","8"],["15","60","45"],["56","336","504","224"]],"row_sums_preview":["0","1","12","120","1120","10080","88704","768768"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a98fae1bf9eeb913","steps":["D","\u03b8+1"],"edges":["e-592ee9fddf6f118c","e-f04eb9181edc5444"],"text":"A085880 \u2192 D \u2192 \u03b8+1"},"headline":"A085880 \u2192 D \u2192 \u03b8+1","chain":null}},{"id":"l-e26b9a8c28e780e3","kind":"fingerprint_related","node_a_id":"n-74bbcae8a0010b72","node_b_id":"n-1e3d729fffbceff3","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-74bbcae8a0010b72","other":{"id":"n-74bbcae8a0010b72","name":"one_plus_derivative of shift_x_plus_one of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","first_index":1,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","1"],["3","4","1"]],"card_rows":[["1"],["2","1"],["3","4","1"],["4","9","6","1"]],"row_sums_preview":["1","3","8","20","48","112","256","576"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1bf99ad40f056d33","steps":["basis A007318","D+1"],"edges":["e-279353e75dd63f59","e-9c5f9650cd2a9a40"],"text":"basis-map coefficients of A026729 \u2192 basis A007318 \u2192 D+1"},"headline":"basis-map coefficients of A026729 \u2192 basis A007318 \u2192 D+1","chain":null}},{"id":"l-e459262da7f70f7e","kind":"fingerprint_related","node_a_id":"n-1e3d729fffbceff3","node_b_id":"n-10f9788d2ae8ab73","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-10f9788d2ae8ab73","other":{"id":"n-10f9788d2ae8ab73","name":"one_plus_derivative of Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","2"],["18","24","6"]],"card_rows":[["1"],["4","2"],["18","24","6"],["96","216","144","24"]],"row_sums_preview":["1","6","48","480","5760","80640","1290240","23224320"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-75a6475b3e851dc5","steps":["D+1"],"edges":["e-74da74941c50f5f3"],"text":"A156992 \u2192 D+1"},"headline":"A156992 \u2192 D+1","chain":null}},{"id":"l-e867ebfd4d919f4f","kind":"fingerprint_related","node_a_id":"n-093b643c6b162488","node_b_id":"n-1e3d729fffbceff3","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-093b643c6b162488","other":{"id":"n-093b643c6b162488","name":"theta_plus_one of basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-1","-2"],["1","4","3"]],"card_rows":[["1"],["-1","-2"],["1","4","3"],["-1","-6","-9","-4"]],"row_sums_preview":["1","-3","8","-20","48","-112","256","-576"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-74c1e94669ff21ff","steps":["\u03b8+1"],"edges":["e-dfc204a5cd0fb4b5"],"text":"basis-map coefficients of A113214, shifted \u2192 \u03b8+1"},"headline":"basis-map coefficients of A113214, shifted \u2192 \u03b8+1","chain":null}},{"id":"l-eac17387b6c1c595","kind":"fingerprint_related","node_a_id":"n-0522939642f68085","node_b_id":"n-1e3d729fffbceff3","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-0522939642f68085","other":{"id":"n-0522939642f68085","name":"one_plus_derivative of Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["10","5"],["45","60","15"]],"card_rows":[["1"],["10","5"],["45","60","15"],["140","315","210","35"]],"row_sums_preview":["1","15","120","700","3360","14112","53760","190080"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-79986ab5023c2764","steps":["D+1"],"edges":["e-5b75c1d710330c14"],"text":"A178821 \u2192 D+1"},"headline":"A178821 \u2192 D+1","chain":null}},{"id":"l-f0d615a3ade892e5","kind":"fingerprint_related","node_a_id":"n-f621432a450963c4","node_b_id":"n-1e3d729fffbceff3","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-f621432a450963c4","other":{"id":"n-f621432a450963c4","name":"theta_plus_one of derivative of basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[[],["-1"],["2","4"]],"card_rows":[["-1"],["2","4"],["-3","-12","-9"],["4","24","36","16"]],"row_sums_preview":["0","-1","6","-24","80","-240","672","-1792"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-74c1e94669ff21ff","steps":["D","\u03b8+1"],"edges":["e-3c99f80a26484c6d","e-df60e95ceb1f63a0"],"text":"basis-map coefficients of A113214, shifted \u2192 D \u2192 \u03b8+1"},"headline":"basis-map coefficients of A113214, shifted \u2192 D \u2192 \u03b8+1","chain":null}},{"id":"l-f241e99bee3e66fd","kind":"fingerprint_related","node_a_id":"n-d86f0fa1a60cdcc7","node_b_id":"n-1e3d729fffbceff3","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-d86f0fa1a60cdcc7","other":{"id":"n-d86f0fa1a60cdcc7","name":"theta_plus_one of Exponential Riordan array [x*exp(x),x].","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],["2","4"]],"card_rows":[["1"],["2","4"],["3","12","9"],["4","24","36","16"]],"row_sums_preview":["0","1","6","24","80","240","672","1792"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-308cf60736924b4c","steps":["\u03b8+1"],"edges":["e-0b4ed87df247fda0"],"text":"A216973 \u2192 \u03b8+1"},"headline":"A216973 \u2192 \u03b8+1","chain":null}},{"id":"l-f4c6405a8c959331","kind":"fingerprint_related","node_a_id":"n-3105a078c60afe34","node_b_id":"n-1e3d729fffbceff3","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-3105a078c60afe34","other":{"id":"n-3105a078c60afe34","name":"one_plus_derivative of Triangular array read by rows: T(n,k) = binomial(n,2)*binomial(n,k), n>=0, 0<=k<=n.","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],[],["3","4","1"]],"card_rows":[["3","4","1"],["12","27","18","3"],["30","96","108","48","6"],["60","250","400","300","100","10"]],"row_sums_preview":["0","0","8","60","288","1120","3840","12096"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c8c19eba9d68881f","steps":["\u00d7(\u2212(1/2)n+(1/2)n\u00b2)","D+1"],"edges":["e-ccfad80d3bf960c1","e-1609333cb5439f2e"],"text":"A007318 \u2192 \u00d7(\u2212(1/2)n+(1/2)n\u00b2) \u2192 D+1"},"headline":"A007318 \u2192 \u00d7(\u2212(1/2)n+(1/2)n\u00b2) \u2192 D+1","chain":null}},{"id":"l-f528ca1479f16620","kind":"fingerprint_related","node_a_id":"n-1e3d729fffbceff3","node_b_id":"n-58b8c2486ebca340","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-58b8c2486ebca340","other":{"id":"n-58b8c2486ebca340","name":"one_plus_derivative of basis_map_A013609 of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(n).","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","2"],["36","48","12"]],"card_rows":[["1"],["4","2"],["36","48","12"],["480","1080","720","120"]],"row_sums_preview":["1","6","96","2400","80640","3386880","170311680","9963233280"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c7809537965f400d","steps":["basis A013609","D+1"],"edges":["e-0f3734a094503340","e-43fef5ac1903f709"],"text":"A085881 \u2192 basis A013609 \u2192 D+1"},"headline":"A085881 \u2192 basis A013609 \u2192 D+1","chain":null}},{"id":"l-f6a7a9f60c2be671","kind":"fingerprint_related","node_a_id":"n-009c8f92daa123a1","node_b_id":"n-1e3d729fffbceff3","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-009c8f92daa123a1","other":{"id":"n-009c8f92daa123a1","name":"theta_plus_one of Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","8"],["10","40","30"]],"card_rows":[["1"],["4","8"],["10","40","30"],["20","120","180","80"]],"row_sums_preview":["1","12","80","400","1680","6272","21504","69120"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a820c1859f68e71c","steps":["\u03b8+1"],"edges":["e-c68d4e8c3af46596"],"text":"A178820 \u2192 \u03b8+1"},"headline":"A178820 \u2192 \u03b8+1","chain":null}},{"id":"l-f8e8291e33ff748d","kind":"fingerprint_related","node_a_id":"n-4fa978d937c48884","node_b_id":"n-1e3d729fffbceff3","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-4fa978d937c48884","other":{"id":"n-4fa978d937c48884","name":"one_plus_derivative of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["2"],["4","2"],["6","8","2"]],"card_rows":[["2"],["4","2"],["6","8","2"],["8","18","12","2"]],"row_sums_preview":["2","6","16","40","96","224","512","1152"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-089a9b4f824a1c20","steps":["D+1"],"edges":["e-33db124cf6524b81"],"text":"A028326 \u2192 D+1"},"headline":"A028326 \u2192 D+1","chain":null}},{"id":"l-fa10a5cfbf3c8bd7","kind":"fingerprint_related","node_a_id":"n-b8376f11b465c0c6","node_b_id":"n-1e3d729fffbceff3","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-b8376f11b465c0c6","other":{"id":"n-b8376f11b465c0c6","name":"one_plus_derivative of derivative of Triangle whose (i,j)-th entry is binomial(i,j)*2^i.","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["2"],["16","8"]],"card_rows":[["2"],["16","8"],["72","96","24"],["256","576","384","64"]],"row_sums_preview":["0","2","24","192","1280","7680","43008","229376"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d28ad9841fc73317","steps":["D","D+1"],"edges":["e-a49f8ff31c907f66","e-14433804f06de5ae"],"text":"A038208 \u2192 D \u2192 D+1"},"headline":"A038208 \u2192 D \u2192 D+1","chain":null}},{"id":"l-fb279f1473e009e1","kind":"fingerprint_related","node_a_id":"n-c84c6091c7207354","node_b_id":"n-1e3d729fffbceff3","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-c84c6091c7207354","other":{"id":"n-c84c6091c7207354","name":"shift_x_plus_one of one_plus_derivative of 3n + 1 preceded by n zeros.","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","4"],["21","28","7"]],"card_rows":[["1"],["8","4"],["21","28","7"],["40","90","60","10"]],"row_sums_preview":["1","12","56","200","624","1792","4864","12672"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ccc86c3dcf2ac685","steps":["D+1","basis A007318"],"edges":["e-133ff869421f85a2","e-427005b54e22a3c7"],"text":"A131431 \u2192 D+1 \u2192 basis A007318"},"headline":"A131431 \u2192 D+1 \u2192 basis A007318","chain":null}},{"id":"l-fc759fdfcb426ca1","kind":"fingerprint_related","node_a_id":"n-eca25272297d9e71","node_b_id":"n-1e3d729fffbceff3","fingerprint":"f6-e99c3824c4864868","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e99c3824c4864868"},"other_id":"n-eca25272297d9e71","other":{"id":"n-eca25272297d9e71","name":"one_plus_derivative of Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*3^j.","first_index":0,"fingerprint":"f6-e99c3824c4864868","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6","3"],["27","36","9"]],"card_rows":[["1"],["6","3"],["27","36","9"],["108","243","162","27"]],"row_sums_preview":["1","9","72","540","3888","27216","186624","1259712"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8d88ef9de74b305f","steps":["D+1"],"edges":["e-8db68d55e3b2a646"],"text":"A038221 \u2192 D+1"},"headline":"A038221 \u2192 D+1","chain":null}}],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:      3;\n  1:     18,       9;\n  2:     81,     108,       27;\n  3:    324,     729,      486,       81;\n  4:   1215,    3888,     4374,     1944,      243;\n  5:   4374,   18225,    29160,    21870,     7290,      729;\n  6:  15309,   78732,   164025,   174960,    98415,    26244,     2187;\n  7:  52488,  321489,   826686,  1148175,   918540,   413343,    91854,    6561;\n  8: 177147, 1259712,  3857868,  6613488,  6889050,  4408992,  1653372,  314928,   19683;\n  9: 590490, 4782969, 17006112, 34720812, 44641044, 37200870, 19840464, 6377292, 1062882, 59049;\n","recurrence_sentence":"The row polynomials satisfy (n+x)*P(n,x) = (3*n+3+(3*n+6)*x+3*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 3.","mathematica":"P[0, x_] := 3;\nP[n_, x_] := P[n, x] = Expand[Together[((3*n + 3 + (3*n + 6)*x + 3*x^2)*P[n - 1, x])/(n + x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 3; for(n = 1, N, my(i = n + 1); v[i] = ((3*n + 3 + (3*n + 6)*x + 3*x^2)*v[i - 1])/(n + x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-51b4f5e7901c2b50","steps":["basis A013610","D+1"],"edges":["e-3ea97f7a94318269","e-1a75bdb00b3f42e8"],"text":"A120910 \u2192 basis A013610 \u2192 D+1"},"headline":"A120910 \u2192 basis A013610 \u2192 D+1","chain":null}
