{"id":"n-8c3fe1c3189f5b0e","revision":1,"first_index":0,"fingerprint":"f6-e99c3824c4864868","name":"one_plus_derivative of Triangle read by rows: T(n,k) = 2^floor(n/2)*binomial(n-1,k-1).","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":[["1"],["4","2"],["6","8","2"],["16","36","24","4"],["20","64","72","32","4"],["48","200","320","240","80","8"],["56","288","600","640","360","96","8"],["128","784","2016","2800","2240","1008","224","16"],["144","1024","3136","5376","5600","3584","1344","256","16"],["320","2592","9216","18816","24192","20160","10752","3456","576","32"],["352","3200","12960","30720","47040","48384","33600","15360","4320","640","32"],["768","7744","35200","95040","168960","206976","177408","105600","42240","10560","1408","64"],["832","9216","46464","140800","285120","405504","413952","304128","158400","56320","12672","1536","64"],["1792","21632","119808","402688","915200","1482624","1757184","1537536","988416","457600","146432","29952","3328","128"],["1920","25088","151424","559104","1409408","2562560","3459456","3514368","2690688","1537536","640640","186368","34944","3584","128"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","6","16","80","192","896","2048","9216","20480","90112","196608","851968","1835008","7864320","16777216"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["4","2"]],"recurrence":{"denominator":{"coeffs":[["-1","1"],["1","0"]]},"terms":[{"coeff":{"coeffs":[["2","6","6","2"],["2","4","2","0"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":163826,"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-40cfcf3b0ff2ebc3"]},{"id":163827,"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-40cfcf3b0ff2ebc3"]},{"id":163828,"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-40cfcf3b0ff2ebc3"]}],"region_exact":{},"incoming_edges":[{"id":"e-40cfcf3b0ff2ebc3","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-8c3fe1c3189f5b0e","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":"e114bd106c641a27653e0dee06de79ca364385cf729691f11cfc7f0a1ee78628","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-a624e5aba8e1c39e","role":"sequence","name":"Triangle read by rows: T(n,k) = 2^floor(n/2)*binomial(n-1,k-1).","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-06323fb6ff6cb8df","kind":"fingerprint_related","node_a_id":"n-8c3fe1c3189f5b0e","node_b_id":"n-3c5dcbc726f4972e","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-0d3f45f6017491de","kind":"fingerprint_related","node_a_id":"n-8c3fe1c3189f5b0e","node_b_id":"n-e78214c646c97ace","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-152d61cc37dd8efd","kind":"fingerprint_related","node_a_id":"n-8c3fe1c3189f5b0e","node_b_id":"n-0522939642f68085","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-1c3068fcfc6385f2","kind":"fingerprint_related","node_a_id":"n-8c3fe1c3189f5b0e","node_b_id":"n-a9e082eabd70d305","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-c8c19eba9d68881f","steps":["\u00d7(1+2n+n\u00b2)","\u03b8+1"],"edges":["e-b78deffc64bc1942","e-d99983198f1cc02c"],"text":"A007318 \u2192 \u00d7(1+2n+n\u00b2) \u2192 \u03b8+1"},"headline":"A007318 \u2192 \u00d7(1+2n+n\u00b2) \u2192 \u03b8+1","chain":null}},{"id":"l-23351b841864c328","kind":"fingerprint_related","node_a_id":"n-8c3fe1c3189f5b0e","node_b_id":"n-da71e01126b2b035","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-2f020583853145bb","kind":"fingerprint_related","node_a_id":"n-0040332d7acdddb4","node_b_id":"n-8c3fe1c3189f5b0e","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-38a3e23f23b20c24","kind":"fingerprint_related","node_a_id":"n-8c3fe1c3189f5b0e","node_b_id":"n-777049cbeb01df27","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-3c018e3184b9322b","kind":"fingerprint_related","node_a_id":"n-8c3fe1c3189f5b0e","node_b_id":"n-1abb04c9738b703c","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-7c03d34c1ae52548"],"text":"A085880 \u2192 \u03b8+1"},"headline":"A085880 \u2192 \u03b8+1","chain":null}},{"id":"l-466e9ac1b756a081","kind":"fingerprint_related","node_a_id":"n-801895951aa20f29","node_b_id":"n-8c3fe1c3189f5b0e","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]"],"region_exact":{},"oeis_ids":["A103406"],"derivation":{"root":"n-c8c19eba9d68881f","steps":["\u03b8+1"],"edges":["e-a112169ce63438eb"],"text":"A007318 \u2192 \u03b8+1"},"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":"A007318 \u2192 \u03b8+1"}},{"id":"l-549f2700f676e9b9","kind":"fingerprint_related","node_a_id":"n-8c3fe1c3189f5b0e","node_b_id":"n-3105a078c60afe34","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-ad66422ab774d8ff"],"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-5ef5403dd096b7bd","kind":"fingerprint_related","node_a_id":"n-8c3fe1c3189f5b0e","node_b_id":"n-009c8f92daa123a1","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-5ef9b40d9a67ca07","kind":"fingerprint_related","node_a_id":"n-4fa978d937c48884","node_b_id":"n-8c3fe1c3189f5b0e","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-673af081df5ac83f","kind":"fingerprint_related","node_a_id":"n-922a52ffa8f2e057","node_b_id":"n-8c3fe1c3189f5b0e","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]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c8c19eba9d68881f","steps":["D","D+1"],"edges":["e-e846abf2e486a264","e-9a80f6183c7ede37"],"text":"A007318 \u2192 D \u2192 D+1"},"headline":"A007318 \u2192 D \u2192 D+1","chain":null}},{"id":"l-674d900ea9456a54","kind":"fingerprint_related","node_a_id":"n-8c3fe1c3189f5b0e","node_b_id":"n-a00a29c17e5648c7","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-6de680d72259ceb1","kind":"fingerprint_related","node_a_id":"n-093b643c6b162488","node_b_id":"n-8c3fe1c3189f5b0e","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-7538b79a907870ae","kind":"fingerprint_related","node_a_id":"n-8c3fe1c3189f5b0e","node_b_id":"n-65d62a2c475a7137","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-75c3aa9cb0c818cf","kind":"fingerprint_related","node_a_id":"n-ac5c1e9a0fbadb48","node_b_id":"n-8c3fe1c3189f5b0e","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-7884b13e8e28aba3","kind":"fingerprint_related","node_a_id":"n-8c3fe1c3189f5b0e","node_b_id":"n-751f2fc33fb80fef","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-c8c19eba9d68881f","steps":["\u00d7(1+(3/2)n+(1/2)n\u00b2)","\u03b8+1"],"edges":["e-739fcf83152eb2a2","e-df5c3bb3c9c4e6f1"],"text":"A007318 \u2192 \u00d7(1+(3/2)n+(1/2)n\u00b2) \u2192 \u03b8+1"},"headline":"A007318 \u2192 \u00d7(1+(3/2)n+(1/2)n\u00b2) \u2192 \u03b8+1","chain":null}},{"id":"l-82827efbef9a488a","kind":"fingerprint_related","node_a_id":"n-8c3fe1c3189f5b0e","node_b_id":"n-c4d59c13bd5c1d27","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-888aeb1964581311","kind":"fingerprint_related","node_a_id":"n-8c3fe1c3189f5b0e","node_b_id":"n-609aa611c94bbe52","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-92c22f7a9757d819","kind":"fingerprint_related","node_a_id":"n-8c3fe1c3189f5b0e","node_b_id":"n-924179b433840212","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-d8e82314ed3cc2a2"],"text":"A178822 \u2192 \u03b8+1"},"headline":"A178822 \u2192 \u03b8+1","chain":null}},{"id":"l-b04bb98db469efa8","kind":"fingerprint_related","node_a_id":"n-f24dc5f77f658331","node_b_id":"n-8c3fe1c3189f5b0e","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]"],"region_exact":{},"oeis_ids":["A093375"],"derivation":{"root":"n-c8c19eba9d68881f","steps":["D+1"],"edges":["e-5c65a1a70531c253"],"text":"A007318 \u2192 D+1"},"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":"A007318 \u2192 D+1"}},{"id":"l-b42280257eb29fc1","kind":"fingerprint_related","node_a_id":"n-8c3fe1c3189f5b0e","node_b_id":"n-925258ea4d4afe84","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-bb4ffd4088181917","kind":"fingerprint_related","node_a_id":"n-d86f0fa1a60cdcc7","node_b_id":"n-8c3fe1c3189f5b0e","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-c8c19eba9d68881f","steps":["D","\u03b8+1"],"edges":["e-e846abf2e486a264","e-0b4ed87df247fda0"],"text":"A007318 \u2192 D \u2192 \u03b8+1"},"headline":"A007318 \u2192 D \u2192 \u03b8+1","chain":null}},{"id":"l-c3fb52b34974e29f","kind":"fingerprint_related","node_a_id":"n-8c3fe1c3189f5b0e","node_b_id":"n-6e023f035bca0041","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-f356f9ef0754fe75"],"text":"A178820 \u2192 D+1"},"headline":"A178820 \u2192 D+1","chain":null}},{"id":"l-cf060c33b06bc983","kind":"fingerprint_related","node_a_id":"n-8c3fe1c3189f5b0e","node_b_id":"n-441d800dafd42935","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]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c8c19eba9d68881f","steps":["\u00d7(1+2n+n\u00b2)","D+1"],"edges":["e-b78deffc64bc1942","e-8324da2f35d5f52e"],"text":"A007318 \u2192 \u00d7(1+2n+n\u00b2) \u2192 D+1"},"headline":"A007318 \u2192 \u00d7(1+2n+n\u00b2) \u2192 D+1","chain":null}},{"id":"l-d329c068feb08cd4","kind":"fingerprint_related","node_a_id":"n-8c3fe1c3189f5b0e","node_b_id":"n-7a1376d2a2b6b46f","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-c8c19eba9d68881f","steps":["\u00d7(1+(3/2)n+(1/2)n\u00b2)","D+1"],"edges":["e-739fcf83152eb2a2","e-1312db2a2710f639"],"text":"A007318 \u2192 \u00d7(1+(3/2)n+(1/2)n\u00b2) \u2192 D+1"},"headline":"A007318 \u2192 \u00d7(1+(3/2)n+(1/2)n\u00b2) \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:   1;\n  1:   4,    2;\n  2:   6,    8,    2;\n  3:  16,   36,   24,     4;\n  4:  20,   64,   72,    32,     4;\n  5:  48,  200,  320,   240,    80,     8;\n  6:  56,  288,  600,   640,   360,    96,     8;\n  7: 128,  784, 2016,  2800,  2240,  1008,   224,   16;\n  8: 144, 1024, 3136,  5376,  5600,  3584,  1344,  256,  16;\n  9: 320, 2592, 9216, 18816, 24192, 20160, 10752, 3456, 576, 32;\n","recurrence_sentence":"The row polynomials satisfy (n-1+x)*P(n,x) = (2*n+2+(4*n+6)*x+(2*n+6)*x^2+2*x^3)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n + 2 + (4*n + 6)*x + (2*n + 6)*x^2 + 2*x^3)*P[n - 2, x])/(n - 1 + x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n + 2 + (4*n + 6)*x + (2*n + 6)*x^2 + 2*x^3)*v[i - 2])/(n - 1 + 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-a624e5aba8e1c39e","steps":["D+1"],"edges":["e-40cfcf3b0ff2ebc3"],"text":"A135838 \u2192 D+1"},"headline":"A135838 \u2192 D+1","chain":null}
