{"id":"n-aa18ad1625904333","revision":1,"first_index":0,"fingerprint":"f6-11aec07ce60b6d61","name":"mobius_x_over_x_plus_one of A127733 * A007318 as infinite lower triangular matrices.","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","8"],["9","36","36"],["16","96","192","128"],["25","200","600","800","400"],["36","360","1440","2880","2880","1152"],["49","588","2940","7840","11760","9408","3136"],["64","896","5376","17920","35840","43008","28672","8192"],["81","1296","9072","36288","90720","145152","145152","82944","20736"],["100","1800","14400","67200","201600","403200","537600","460800","230400","51200"],["121","2420","21780","116160","406560","975744","1626240","1858560","1393920","619520","123904"],["144","3168","31680","190080","760320","2128896","4257792","6082560","6082560","4055040","1622016","294912"],["169","4056","44616","297440","1338480","4283136","9993984","17132544","21415680","19036160","11421696","4153344","692224"],["196","5096","61152","448448","2242240","8072064","21525504","43051008","64576512","71751680","57401344","31309824","10436608","1605632"],["225","6300","81900","655200","3603600","14414400","43243200","98841600","172972800","230630400","230630400","167731200","83865600","25804800","3686400"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","12","81","432","2025","8748","35721","139968","531441","1968300","7144929","25509168","89813529","312487308","1076168025"],"nonn":true,"definitions":[{"position":0,"source":"derived:mobius_x_over_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0","0"],["0","0"],["1","1"]]},"terms":[{"coeff":{"coeffs":[["1","3","2"],["2","6","4"],["1","3","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":164241,"position":0,"claim":"real_rooted","from_index":0,"region_key":"[-1, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in [-1, 0]","region_exact":null,"edges":["e-963b3afafd9f7f70"]},{"id":164242,"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-963b3afafd9f7f70"]},{"id":164243,"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-963b3afafd9f7f70"]}],"region_exact":{},"incoming_edges":[{"id":"e-963b3afafd9f7f70","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"mobius\",\"a\":\"1\",\"b\":\"0\",\"c\":\"1\",\"e\":\"1\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"},"operation_detail":"mobius, shift 0","conclusion_id":"n-aa18ad1625904333","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":"fTransform","lean_declarations":["RealRooted.splits_fPolynomial"],"certificate_hash":"abb365afa69900d6cef3653bbe3c6c42a1e36ff7e099e52043c838af12463c9e","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-375ba152c4aebf61","role":"sequence","name":"A127733 * A007318 as infinite lower triangular matrices.","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-022f1eac3651c863","kind":"fingerprint_related","node_a_id":"n-aa18ad1625904333","node_b_id":"n-8f0d536701873284","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-8f0d536701873284","other":{"id":"n-8f0d536701873284","name":"basis_map_A013611 of Triangle read by rows: T(n,k) = 2^floor(n/2)*binomial(n-1,k-1).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","8"],["8","32","32"]],"card_rows":[["1"],["4","8"],["8","32","32"],["32","192","384","256"]],"row_sums_preview":["1","12","72","864","5184","62208","373248","4478976"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a624e5aba8e1c39e","steps":["basis A013611"],"edges":["e-0f9b3c3168d8e647"],"text":"A135838 \u2192 basis A013611"},"headline":"A135838 \u2192 basis A013611","chain":null}},{"id":"l-042db3e673a95da0","kind":"fingerprint_related","node_a_id":"n-17166a4ad91393c8","node_b_id":"n-aa18ad1625904333","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-17166a4ad91393c8","other":{"id":"n-17166a4ad91393c8","name":"basis_map_A007318 of Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","1"],["8","8","2"]],"card_rows":[["1"],["2","1"],["8","8","2"],["48","72","36","6"]],"row_sums_preview":["1","3","18","162","1944","29160","524880","11022480"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cb5d49e71e46f2ed","steps":["basis A007318"],"edges":["e-703e907af1aa9666"],"text":"A196347 \u2192 basis A007318"},"headline":"A196347 \u2192 basis A007318","chain":null}},{"id":"l-0629252a6ab8cff1","kind":"fingerprint_related","node_a_id":"n-416d26cbd480eee5","node_b_id":"n-aa18ad1625904333","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-416d26cbd480eee5","other":{"id":"n-416d26cbd480eee5","name":"shift_x_plus_one of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["2"],["4","2"],["8","8","2"]],"card_rows":[["2"],["4","2"],["8","8","2"],["16","24","12","2"]],"row_sums_preview":["2","6","18","54","162","486","1458","4374"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-089a9b4f824a1c20","steps":["basis A007318"],"edges":["e-014dd8a0f9594ad5"],"text":"A028326 \u2192 basis A007318"},"headline":"A028326 \u2192 basis A007318","chain":null}},{"id":"l-083fac1417f079d7","kind":"fingerprint_related","node_a_id":"n-aa18ad1625904333","node_b_id":"n-0be19de8f4a58a03","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-0be19de8f4a58a03","other":{"id":"n-0be19de8f4a58a03","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*2^i.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","8"],["16","64","64"]],"card_rows":[["1"],["4","8"],["16","64","64"],["64","384","768","512"]],"row_sums_preview":["1","12","144","1728","20736","248832","2985984","35831808"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d28ad9841fc73317","steps":["basis A013611"],"edges":["e-e6fcabb7b5383fc4"],"text":"A038208 \u2192 basis A013611"},"headline":"A038208 \u2192 basis A013611","chain":null}},{"id":"l-10ab18170a4d2001","kind":"fingerprint_related","node_a_id":"n-03ac40ba4fd62e4c","node_b_id":"n-aa18ad1625904333","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-03ac40ba4fd62e4c","other":{"id":"n-03ac40ba4fd62e4c","name":"basis_map_A007318 of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(n).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","1"],["12","12","3"]],"card_rows":[["1"],["2","1"],["12","12","3"],["120","180","90","15"]],"row_sums_preview":["1","3","27","405","8505","229635","7577955","295540245"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c7809537965f400d","steps":["basis A007318"],"edges":["e-5048ce3ebd740a6f"],"text":"A085881 \u2192 basis A007318"},"headline":"A085881 \u2192 basis A007318","chain":null}},{"id":"l-10ae648ebd96bcfa","kind":"fingerprint_related","node_a_id":"n-7f05fd9e9c6948d8","node_b_id":"n-aa18ad1625904333","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-7f05fd9e9c6948d8","other":{"id":"n-7f05fd9e9c6948d8","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*3^j.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6","3"],["36","36","9"]],"card_rows":[["1"],["6","3"],["36","36","9"],["216","324","162","27"]],"row_sums_preview":["1","9","81","729","6561","59049","531441","4782969"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7b91a7a037fb3513","steps":["basis A013610"],"edges":["e-2d1052dd37eda11b"],"text":"A038243 \u2192 basis A013610"},"headline":"A038243 \u2192 basis A013610","chain":null}},{"id":"l-10f30631e7cd6b41","kind":"fingerprint_related","node_a_id":"n-5d91f73259c0b712","node_b_id":"n-aa18ad1625904333","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-5d91f73259c0b712","other":{"id":"n-5d91f73259c0b712","name":"basis_map_A007318 of Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","4"],["40","40","10"]],"card_rows":[["1"],["8","4"],["40","40","10"],["160","240","120","20"]],"row_sums_preview":["1","12","90","540","2835","13608","61236","262440"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a820c1859f68e71c","steps":["basis A007318"],"edges":["e-78029e34ed2e59f8"],"text":"A178820 \u2192 basis A007318"},"headline":"A178820 \u2192 basis A007318","chain":null}},{"id":"l-1888942b67e98689","kind":"fingerprint_related","node_a_id":"n-076de49b14800916","node_b_id":"n-aa18ad1625904333","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-076de49b14800916","other":{"id":"n-076de49b14800916","name":"derivative of Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs (i.e., subwords of maximal length) of identical letters (1 <= k <= n).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["6"],["12","24"]],"card_rows":[["6"],["12","24"],["18","72","72"],["24","144","288","192"]],"row_sums_preview":["0","6","36","162","648","2430","8748","30618"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-765f9e2b013b3c68","steps":["D"],"edges":["e-13c4b00cf881ee86"],"text":"A120909 \u2192 D"},"headline":"A120909 \u2192 D","chain":null}},{"id":"l-1d5091ae727cb441","kind":"fingerprint_related","node_a_id":"n-0b43db26e266f660","node_b_id":"n-aa18ad1625904333","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-0b43db26e266f660","other":{"id":"n-0b43db26e266f660","name":"basis_map_A007318 of Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","2"],["24","24","6"]],"card_rows":[["1"],["4","2"],["24","24","6"],["160","240","120","20"]],"row_sums_preview":["1","6","54","540","5670","61236","673596","7505784"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-dd77bf712b1df996","steps":["basis A007318"],"edges":["e-ced3e67c3d44f436"],"text":"A356546 \u2192 basis A007318"},"headline":"A356546 \u2192 basis A007318","chain":null}},{"id":"l-1e018389d3f23d70","kind":"fingerprint_related","node_a_id":"n-51b4f5e7901c2b50","node_b_id":"n-aa18ad1625904333","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-51b4f5e7901c2b50","other":{"id":"n-51b4f5e7901c2b50","name":"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":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["3"],["6","3"],["12","12","3"]],"card_rows":[["3"],["6","3"],["12","12","3"],["24","36","18","3"]],"row_sums_preview":["3","9","27","81","243","729","2187","6561"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":["A120910"],"derivation":{"root":"n-51b4f5e7901c2b50","steps":[],"edges":[],"text":"A120910"},"headline":"A120910: 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.","chain":null}},{"id":"l-1e1f40aa435dd9d8","kind":"fingerprint_related","node_a_id":"n-3b6ec7300891f6df","node_b_id":"n-aa18ad1625904333","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-3b6ec7300891f6df","other":{"id":"n-3b6ec7300891f6df","name":"mobius_x_over_x_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":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","6"],["6","24","24"]],"card_rows":[["1"],["3","6"],["6","24","24"],["10","60","120","80"]],"row_sums_preview":["1","9","54","270","1215","5103","20412","78732"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c8c19eba9d68881f","steps":["\u00d7(1+(3/2)n+(1/2)n\u00b2)","x \u21a6 x/(x+1)"],"edges":["e-739fcf83152eb2a2","e-53cd3f44d77a4ed8"],"text":"A007318 \u2192 \u00d7(1+(3/2)n+(1/2)n\u00b2) \u2192 x \u21a6 x/(x+1)"},"headline":"A007318 \u2192 \u00d7(1+(3/2)n+(1/2)n\u00b2) \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-1e6b002acae20683","kind":"fingerprint_related","node_a_id":"n-af754f07daf65b5e","node_b_id":"n-aa18ad1625904333","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-af754f07daf65b5e","other":{"id":"n-af754f07daf65b5e","name":"Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","1"],["4","4","1"]],"card_rows":[["1"],["2","1"],["4","4","1"],["8","12","6","1"]],"row_sums_preview":["1","3","9","27","81","243","729","2187"],"regions":["(-inf, 0)","(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":["A038207"],"derivation":{"root":"n-af754f07daf65b5e","steps":[],"edges":[],"text":"A038207"},"headline":"A038207: Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).","chain":null}},{"id":"l-27554e776b5e1201","kind":"fingerprint_related","node_a_id":"n-aa18ad1625904333","node_b_id":"n-9805b07a338913cb","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-9805b07a338913cb","other":{"id":"n-9805b07a338913cb","name":"basis_map_A013611 of Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*11^j.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["22","44"],["484","1936","1936"]],"card_rows":[["1"],["22","44"],["484","1936","1936"],["10648","63888","127776","85184"]],"row_sums_preview":["1","66","4356","287496","18974736","1252332576","82653950016","5455160701056"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8064e72cf80883a7","steps":["basis A013611"],"edges":["e-6a2839a9935a683b"],"text":"A038325 \u2192 basis A013611"},"headline":"A038325 \u2192 basis A013611","chain":null}},{"id":"l-287ec0d37c171ecb","kind":"fingerprint_related","node_a_id":"n-7c4e320f16a7ccb1","node_b_id":"n-aa18ad1625904333","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-7c4e320f16a7ccb1","other":{"id":"n-7c4e320f16a7ccb1","name":"shift_x_plus_one of Exponential Riordan array [x*exp(x),x].","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","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"],["12","12","3"],["32","48","24","4"]],"row_sums_preview":["0","1","6","27","108","405","1458","5103"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-af754f07daf65b5e","steps":["D"],"edges":["e-fd845f603f8fe831"],"text":"A038207 \u2192 D"},"headline":"A038207 \u2192 D","chain":null}},{"id":"l-2f5d1c64e8be6208","kind":"fingerprint_related","node_a_id":"n-aa18ad1625904333","node_b_id":"n-986a623282b41276","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-986a623282b41276","other":{"id":"n-986a623282b41276","name":"basis_map_A013609 of basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).","first_index":1,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-1","-2"],["2","8","8"]],"card_rows":[["1"],["-1","-2"],["2","8","8"],["-6","-36","-72","-48"]],"row_sums_preview":["1","-3","18","-162","1944","-29160","524880","-11022480"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cf1765de6ac2bd4b","steps":["basis A013609"],"edges":["e-fbc81b67b4fab4f6"],"text":"basis-map coefficients of A021012 \u2192 basis A013609"},"headline":"basis-map coefficients of A021012 \u2192 basis A013609","chain":null}},{"id":"l-334267f0fb225965","kind":"fingerprint_related","node_a_id":"n-aa18ad1625904333","node_b_id":"n-a67bbce5c2713a1e","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-a67bbce5c2713a1e","other":{"id":"n-a67bbce5c2713a1e","name":"basis_map_A013611 of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["2"],["4","8"],["8","32","32"]],"card_rows":[["2"],["4","8"],["8","32","32"],["16","96","192","128"]],"row_sums_preview":["2","12","72","432","2592","15552","93312","559872"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-089a9b4f824a1c20","steps":["basis A013611"],"edges":["e-3deac81ab612cd2e"],"text":"A028326 \u2192 basis A013611"},"headline":"A028326 \u2192 basis A013611","chain":null}},{"id":"l-3362dc17a85e7a4b","kind":"fingerprint_related","node_a_id":"n-8d4637810467d09a","node_b_id":"n-aa18ad1625904333","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-8d4637810467d09a","other":{"id":"n-8d4637810467d09a","name":"basis_map_A007318 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":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","1"],["8","8","2"]],"card_rows":[["1"],["2","1"],["8","8","2"],["40","60","30","5"]],"row_sums_preview":["1","3","18","135","1134","10206","96228","938223"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a98fae1bf9eeb913","steps":["x \u21a6 x+1"],"edges":["e-48e55eb2c8b6f3ba"],"text":"A085880 \u2192 x \u21a6 x+1"},"headline":"A085880 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-3a2400f6c012b713","kind":"fingerprint_related","node_a_id":"n-aa18ad1625904333","node_b_id":"n-439ac61e3f209ac0","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-439ac61e3f209ac0","other":{"id":"n-439ac61e3f209ac0","name":"basis_map_A013609 of Triangle read by rows: row n consists of n zeros followed by n!.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","2"],["2","8","8"]],"card_rows":[["1"],["1","2"],["2","8","8"],["6","36","72","48"]],"row_sums_preview":["1","3","18","162","1944","29160","524880","11022480"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-fbb4b9430d9de872","steps":["basis A013609"],"edges":["e-eea0c95a12d0633a"],"text":"A136572 \u2192 basis A013609"},"headline":"A136572 \u2192 basis A013609","chain":null}},{"id":"l-3a65e68928a51ba0","kind":"fingerprint_related","node_a_id":"n-aa18ad1625904333","node_b_id":"n-b4b87865f0ed6035","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-b4b87865f0ed6035","other":{"id":"n-b4b87865f0ed6035","name":"basis_map_A013611 of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(n).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","4"],["12","48","48"]],"card_rows":[["1"],["2","4"],["12","48","48"],["120","720","1440","960"]],"row_sums_preview":["1","6","108","3240","136080","7348320","484989120","37829151360"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c7809537965f400d","steps":["basis A013611"],"edges":["e-20dcbad557ba463b"],"text":"A085881 \u2192 basis A013611"},"headline":"A085881 \u2192 basis A013611","chain":null}},{"id":"l-434d1ebbc352dac8","kind":"fingerprint_related","node_a_id":"n-aa18ad1625904333","node_b_id":"n-53ad300b8c449980","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-53ad300b8c449980","other":{"id":"n-53ad300b8c449980","name":"basis_map_A013610 of Triangle of coefficients in expansion of (1+2*x)^n.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","6"],["9","36","36"]],"card_rows":[["1"],["3","6"],["9","36","36"],["27","162","324","216"]],"row_sums_preview":["1","9","81","729","6561","59049","531441","4782969"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-71965e79e47a4526","steps":["basis A013610"],"edges":["e-14a3a792ab5fe5af"],"text":"A013609 \u2192 basis A013610"},"headline":"A013609 \u2192 basis A013610","chain":null}},{"id":"l-4a0d24abf84e8099","kind":"fingerprint_related","node_a_id":"n-aa18ad1625904333","node_b_id":"n-e69b830457b341cf","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-e69b830457b341cf","other":{"id":"n-e69b830457b341cf","name":"basis_map_A013611 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":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","4"],["8","32","32"]],"card_rows":[["1"],["2","4"],["8","32","32"],["40","240","480","320"]],"row_sums_preview":["1","6","72","1080","18144","326592","6158592","120092544"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a98fae1bf9eeb913","steps":["basis A013611"],"edges":["e-b498323f12f05355"],"text":"A085880 \u2192 basis A013611"},"headline":"A085880 \u2192 basis A013611","chain":null}},{"id":"l-500419eab23211dd","kind":"fingerprint_related","node_a_id":"n-aa18ad1625904333","node_b_id":"n-1e156f12069736a3","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-1e156f12069736a3","other":{"id":"n-1e156f12069736a3","name":"basis_map_A013611 of basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-2","-4"],["4","16","16"]],"card_rows":[["1"],["-2","-4"],["4","16","16"],["-8","-48","-96","-64"]],"row_sums_preview":["1","-6","36","-216","1296","-7776","46656","-279936"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-74c1e94669ff21ff","steps":["basis A013611"],"edges":["e-6ef42593efb6357c"],"text":"basis-map coefficients of A113214, shifted \u2192 basis A013611"},"headline":"basis-map coefficients of A113214, shifted \u2192 basis A013611","chain":null}},{"id":"l-50202f260fecbdef","kind":"fingerprint_related","node_a_id":"n-0ea1c8d30da3abae","node_b_id":"n-aa18ad1625904333","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-0ea1c8d30da3abae","other":{"id":"n-0ea1c8d30da3abae","name":"mobius_x_over_x_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":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],[],["1","4","4"]],"card_rows":[["1","4","4"],["3","18","36","24"],["6","48","144","192","96"],["10","100","400","800","800","320"]],"row_sums_preview":["0","0","9","81","486","2430","10935","45927"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c8c19eba9d68881f","steps":["\u00d7(\u2212(1/2)n+(1/2)n\u00b2)","x \u21a6 x/(x+1)"],"edges":["e-ccfad80d3bf960c1","e-018032959df2ae1c"],"text":"A007318 \u2192 \u00d7(\u2212(1/2)n+(1/2)n\u00b2) \u2192 x \u21a6 x/(x+1)"},"headline":"A007318 \u2192 \u00d7(\u2212(1/2)n+(1/2)n\u00b2) \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-50544917cdf0e011","kind":"fingerprint_related","node_a_id":"n-df44f407d23e429d","node_b_id":"n-aa18ad1625904333","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-df44f407d23e429d","other":{"id":"n-df44f407d23e429d","name":"basis_map_A007318 of Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","2"],["24","24","6"]],"card_rows":[["1"],["4","2"],["24","24","6"],["192","288","144","24"]],"row_sums_preview":["1","6","54","648","9720","174960","3674160","88179840"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-75a6475b3e851dc5","steps":["basis A007318"],"edges":["e-bd56c468ffc76132"],"text":"A156992 \u2192 basis A007318"},"headline":"A156992 \u2192 basis A007318","chain":null}},{"id":"l-55ec0403a9b3a8f7","kind":"fingerprint_related","node_a_id":"n-aa18ad1625904333","node_b_id":"n-3287a7cc800f9f98","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-3287a7cc800f9f98","other":{"id":"n-3287a7cc800f9f98","name":"mobius_x_over_x_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":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["3"],["15","30"],["45","180","180"]],"card_rows":[["3"],["15","30"],["45","180","180"],["105","630","1260","840"]],"row_sums_preview":["3","45","405","2835","17010","91854","459270","2165130"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-dd4aa31f3d11d152","steps":["x \u21a6 x/(x+1)"],"edges":["e-3edca5aad3e7cb78"],"text":"A344565 \u2192 x \u21a6 x/(x+1)"},"headline":"A344565 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-5b1e8ded1aa44742","kind":"fingerprint_related","node_a_id":"n-63d7f3b1eb8cf1f2","node_b_id":"n-aa18ad1625904333","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-63d7f3b1eb8cf1f2","other":{"id":"n-63d7f3b1eb8cf1f2","name":"shift_x_plus_one of A127733 * A007318 as infinite lower triangular matrices.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","4"],["36","36","9"]],"card_rows":[["1"],["8","4"],["36","36","9"],["128","192","96","16"]],"row_sums_preview":["1","12","81","432","2025","8748","35721","139968"],"regions":["(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c8c19eba9d68881f","steps":["\u00d7(1+2n+n\u00b2)","x \u21a6 x+1"],"edges":["e-b78deffc64bc1942","e-18ace392e620d7bd"],"text":"A007318 \u2192 \u00d7(1+2n+n\u00b2) \u2192 x \u21a6 x+1"},"headline":"A007318 \u2192 \u00d7(1+2n+n\u00b2) \u2192 x \u21a6 x+1","chain":null}},{"id":"l-5b38b8eb1fc70b7d","kind":"fingerprint_related","node_a_id":"n-b061091a2d541bea","node_b_id":"n-aa18ad1625904333","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-b061091a2d541bea","other":{"id":"n-b061091a2d541bea","name":"mobius_x_over_x_plus_one of Triangle of denominators in Leibniz's Harmonic Triangle a(n,k), n >= 1, 1 <= k <= n.","first_index":1,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","4"],["3","12","12"]],"card_rows":[["1"],["2","4"],["3","12","12"],["4","24","48","32"]],"row_sums_preview":["1","6","27","108","405","1458","5103","17496"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c8c19eba9d68881f","steps":["D","x \u21a6 x/(x+1)"],"edges":["e-fd16b43297f2bb3e","e-bbc53a65557a79f5"],"text":"A007318 \u2192 D \u2192 x \u21a6 x/(x+1)"},"headline":"A007318 \u2192 D \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-60beac4e645985b8","kind":"fingerprint_related","node_a_id":"n-5a242687ab527719","node_b_id":"n-aa18ad1625904333","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-5a242687ab527719","other":{"id":"n-5a242687ab527719","name":"basis_map_A007318 of Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["12","6"],["84","84","21"]],"card_rows":[["1"],["12","6"],["84","84","21"],["448","672","336","56"]],"row_sums_preview":["1","18","189","1512","10206","61236","336798","1732104"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-97228a88c81561dc","steps":["x \u21a6 x+1"],"edges":["e-a52727d7b51dd683"],"text":"A178822 \u2192 x \u21a6 x+1"},"headline":"A178822 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-649b6926629ecdb6","kind":"fingerprint_related","node_a_id":"n-2db2a1c44af41062","node_b_id":"n-aa18ad1625904333","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-2db2a1c44af41062","other":{"id":"n-2db2a1c44af41062","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["576","576","144"],["13824","20736","10368","1728"],["331776","663552","497664","165888","20736"]],"card_rows":[["576","576","144"],["13824","20736","10368","1728"],["331776","663552","497664","165888","20736"],["7962624","19906560","19906560","9953280","2488320","248832"]],"row_sums_preview":["1296","46656","1679616","60466176","2176782336","78364164096","2821109907456","101559956668416"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c1532953a6736702","steps":["basis A007318"],"edges":["e-a2ff444bbbdc7b83"],"text":"A038338 \u2192 basis A007318"},"headline":"A038338 \u2192 basis A007318","chain":null}},{"id":"l-669153c813dd5aa7","kind":"fingerprint_related","node_a_id":"n-aa18ad1625904333","node_b_id":"n-5317d1468ab69371","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-5317d1468ab69371","other":{"id":"n-5317d1468ab69371","name":"basis_map_A013611 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":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["3"],["30","60"],["180","720","720"]],"card_rows":[["3"],["30","60"],["180","720","720"],["840","5040","10080","6720"]],"row_sums_preview":["3","90","1620","22680","272160","2939328","29393280","277136640"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-dd4aa31f3d11d152","steps":["basis A013611"],"edges":["e-8503bd9580ad59ab"],"text":"A344565 \u2192 basis A013611"},"headline":"A344565 \u2192 basis A013611","chain":null}},{"id":"l-6d9a663c170ef1be","kind":"fingerprint_related","node_a_id":"n-71965e79e47a4526","node_b_id":"n-aa18ad1625904333","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-71965e79e47a4526","other":{"id":"n-71965e79e47a4526","name":"Triangle of coefficients in expansion of (1+2*x)^n.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","2"],["1","4","4"]],"card_rows":[["1"],["1","2"],["1","4","4"],["1","6","12","8"]],"row_sums_preview":["1","3","9","27","81","243","729","2187"],"regions":["(-inf, 0)","(-inf, 0]","[-1, 0]"],"region_exact":{},"oeis_ids":["A013609"],"derivation":{"root":"n-71965e79e47a4526","steps":[],"edges":[],"text":"A013609"},"headline":"A013609: Triangle of coefficients in expansion of (1+2*x)^n.","chain":null}},{"id":"l-78f85fc8152cd2c8","kind":"fingerprint_related","node_a_id":"n-6ba906f5d673fec6","node_b_id":"n-aa18ad1625904333","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-6ba906f5d673fec6","other":{"id":"n-6ba906f5d673fec6","name":"mobius_x_over_x_plus_one of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["2"],["2","4"],["2","8","8"]],"card_rows":[["2"],["2","4"],["2","8","8"],["2","12","24","16"]],"row_sums_preview":["2","6","18","54","162","486","1458","4374"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-089a9b4f824a1c20","steps":["x \u21a6 x/(x+1)"],"edges":["e-71255c0d2ce9c4bc"],"text":"A028326 \u2192 x \u21a6 x/(x+1)"},"headline":"A028326 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-7bde7efc45991ff4","kind":"fingerprint_related","node_a_id":"n-aa18ad1625904333","node_b_id":"n-ed2a02208588b66b","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-ed2a02208588b66b","other":{"id":"n-ed2a02208588b66b","name":"mobius_x_over_x_plus_one of Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6","12"],["21","84","84"]],"card_rows":[["1"],["6","12"],["21","84","84"],["56","336","672","448"]],"row_sums_preview":["1","18","189","1512","10206","61236","336798","1732104"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-97228a88c81561dc","steps":["x \u21a6 x/(x+1)"],"edges":["e-376fcdbb8dac8b5b"],"text":"A178822 \u2192 x \u21a6 x/(x+1)"},"headline":"A178822 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-7db374f90c15a90e","kind":"fingerprint_related","node_a_id":"n-aa18ad1625904333","node_b_id":"n-465e4344a8d94925","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-465e4344a8d94925","other":{"id":"n-465e4344a8d94925","name":"basis_map_A013609 of Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed by the term 2^k. Triangle by columns, (2^k, 0, 0, 0, ...).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","4"],["4","16","16"]],"card_rows":[["1"],["2","4"],["4","16","16"],["8","48","96","64"]],"row_sums_preview":["1","6","36","216","1296","7776","46656","279936"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ae7910bbe46f8a41","steps":["basis A013609"],"edges":["e-6abeab86de871564"],"text":"A130123 \u2192 basis A013609"},"headline":"A130123 \u2192 basis A013609","chain":null}},{"id":"l-8085e0154c422837","kind":"fingerprint_related","node_a_id":"n-e93f0e54dbc33ad5","node_b_id":"n-aa18ad1625904333","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-e93f0e54dbc33ad5","other":{"id":"n-e93f0e54dbc33ad5","name":"basis_map_A007318 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":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["3"],["30","15"],["180","180","45"]],"card_rows":[["3"],["30","15"],["180","180","45"],["840","1260","630","105"]],"row_sums_preview":["3","45","405","2835","17010","91854","459270","2165130"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-dd4aa31f3d11d152","steps":["basis A007318"],"edges":["e-5d53734a383a65e0"],"text":"A344565 \u2192 basis A007318"},"headline":"A344565 \u2192 basis A007318","chain":null}},{"id":"l-874b669a9f766c53","kind":"fingerprint_related","node_a_id":"n-aa18ad1625904333","node_b_id":"n-8f1f80f146f45463","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-8f1f80f146f45463","other":{"id":"n-8f1f80f146f45463","name":"mobius_x_over_x_plus_one of Triangle whose (i,j)-th entry is binomial(i,j)*2^i.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","4"],["4","16","16"]],"card_rows":[["1"],["2","4"],["4","16","16"],["8","48","96","64"]],"row_sums_preview":["1","6","36","216","1296","7776","46656","279936"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d28ad9841fc73317","steps":["x \u21a6 x/(x+1)"],"edges":["e-19854cf600900932"],"text":"A038208 \u2192 x \u21a6 x/(x+1)"},"headline":"A038208 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-8ab9b47627f438db","kind":"fingerprint_related","node_a_id":"n-aa18ad1625904333","node_b_id":"n-219653841046440e","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-219653841046440e","other":{"id":"n-219653841046440e","name":"basis_map_A013610 of Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs (i.e., subwords of maximal length) of identical letters (1 <= k <= n).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["3"],["9","18"],["27","108","108"]],"card_rows":[["3"],["9","18"],["27","108","108"],["81","486","972","648"]],"row_sums_preview":["3","27","243","2187","19683","177147","1594323","14348907"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-765f9e2b013b3c68","steps":["basis A013610"],"edges":["e-284e6c48cfe1aa93"],"text":"A120909 \u2192 basis A013610"},"headline":"A120909 \u2192 basis A013610","chain":null}},{"id":"l-91134d7cef44e8be","kind":"fingerprint_related","node_a_id":"n-aa18ad1625904333","node_b_id":"n-8dc7b8a12ecdf1ca","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-8dc7b8a12ecdf1ca","other":{"id":"n-8dc7b8a12ecdf1ca","name":"mobius_x_over_x_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":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","2"],["2","8","8"]],"card_rows":[["1"],["1","2"],["2","8","8"],["5","30","60","40"]],"row_sums_preview":["1","3","18","135","1134","10206","96228","938223"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a98fae1bf9eeb913","steps":["x \u21a6 x/(x+1)"],"edges":["e-e1834b9d49e58efd"],"text":"A085880 \u2192 x \u21a6 x/(x+1)"},"headline":"A085880 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-962f5bd3379af931","kind":"fingerprint_related","node_a_id":"n-aa18ad1625904333","node_b_id":"n-c80fa23f7424bb55","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-c80fa23f7424bb55","other":{"id":"n-c80fa23f7424bb55","name":"basis_map_A013609 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":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","2"],["1","4","4"]],"card_rows":[["1"],["1","2"],["1","4","4"],["1","6","12","8"]],"row_sums_preview":["1","3","9","27","81","243","729","2187"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1bf99ad40f056d33","steps":["basis A013609"],"edges":["e-7869781294d5f3a8"],"text":"basis-map coefficients of A026729 \u2192 basis A013609"},"headline":"basis-map coefficients of A026729 \u2192 basis A013609","chain":null}},{"id":"l-967d388252fbf695","kind":"fingerprint_related","node_a_id":"n-e8ae0bac7c0efceb","node_b_id":"n-aa18ad1625904333","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-e8ae0bac7c0efceb","other":{"id":"n-e8ae0bac7c0efceb","name":"mobius_x_over_x_plus_one of Triangle read by rows: T(n,k) = 2^floor(n/2)*binomial(n-1,k-1).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","4"],["2","8","8"]],"card_rows":[["1"],["2","4"],["2","8","8"],["4","24","48","32"]],"row_sums_preview":["1","6","18","108","324","1944","5832","34992"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a624e5aba8e1c39e","steps":["x \u21a6 x/(x+1)"],"edges":["e-9f3b1e4aefe8044e"],"text":"A135838 \u2192 x \u21a6 x/(x+1)"},"headline":"A135838 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-99c89f6af57924ea","kind":"fingerprint_related","node_a_id":"n-8f9866a4e131e95d","node_b_id":"n-aa18ad1625904333","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-8f9866a4e131e95d","other":{"id":"n-8f9866a4e131e95d","name":"basis_map_A007318 of Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*11^j.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["22","11"],["484","484","121"]],"card_rows":[["1"],["22","11"],["484","484","121"],["10648","15972","7986","1331"]],"row_sums_preview":["1","33","1089","35937","1185921","39135393","1291467969","42618442977"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8064e72cf80883a7","steps":["basis A007318"],"edges":["e-a5d0659f5315d3b7"],"text":"A038325 \u2192 basis A007318"},"headline":"A038325 \u2192 basis A007318","chain":null}},{"id":"l-9e17d7fbc2f7954c","kind":"fingerprint_related","node_a_id":"n-f8c9dbb903f44fdc","node_b_id":"n-aa18ad1625904333","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-f8c9dbb903f44fdc","other":{"id":"n-f8c9dbb903f44fdc","name":"shift_x_plus_one of basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-2","-1"],["4","4","1"]],"card_rows":[["1"],["-2","-1"],["4","4","1"],["-8","-12","-6","-1"]],"row_sums_preview":["1","-3","9","-27","81","-243","729","-2187"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-74c1e94669ff21ff","steps":["x \u21a6 x+1"],"edges":["e-51a320c1f4e00092"],"text":"basis-map coefficients of A113214, shifted \u2192 x \u21a6 x+1"},"headline":"basis-map coefficients of A113214, shifted \u2192 x \u21a6 x+1","chain":null}},{"id":"l-9f2d015eb1baaa67","kind":"fingerprint_related","node_a_id":"n-765f9e2b013b3c68","node_b_id":"n-aa18ad1625904333","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-765f9e2b013b3c68","other":{"id":"n-765f9e2b013b3c68","name":"Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs (i.e., subwords of maximal length) of identical letters (1 <= k <= n).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["3"],["3","6"],["3","12","12"]],"card_rows":[["3"],["3","6"],["3","12","12"],["3","18","36","24"]],"row_sums_preview":["3","9","27","81","243","729","2187","6561"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":["A120909"],"derivation":{"root":"n-765f9e2b013b3c68","steps":[],"edges":[],"text":"A120909"},"headline":"A120909: Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs (i.e., subwords of maximal length) of identical letters (1 <= k <= n).","chain":null}},{"id":"l-a47064fb3417f1ba","kind":"fingerprint_related","node_a_id":"n-aa18ad1625904333","node_b_id":"n-ce701caf7dd0b8e7","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-ce701caf7dd0b8e7","other":{"id":"n-ce701caf7dd0b8e7","name":"basis_map_A013611 of Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","4"],["8","32","32"]],"card_rows":[["1"],["2","4"],["8","32","32"],["48","288","576","384"]],"row_sums_preview":["1","6","72","1296","31104","933120","33592320","1410877440"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cb5d49e71e46f2ed","steps":["basis A013611"],"edges":["e-f21031c0731e2731"],"text":"A196347 \u2192 basis A013611"},"headline":"A196347 \u2192 basis A013611","chain":null}},{"id":"l-a64cd59f26dc9811","kind":"fingerprint_related","node_a_id":"n-3f5f8a84df712aee","node_b_id":"n-aa18ad1625904333","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-3f5f8a84df712aee","other":{"id":"n-3f5f8a84df712aee","name":"basis_map_A007318 of Triangle read by rows: T(n,k) = 2^floor(n/2)*binomial(n-1,k-1).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","2"],["8","8","2"]],"card_rows":[["1"],["4","2"],["8","8","2"],["32","48","24","4"]],"row_sums_preview":["1","6","18","108","324","1944","5832","34992"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a624e5aba8e1c39e","steps":["x \u21a6 x+1"],"edges":["e-7cade0dc9a919c2f"],"text":"A135838 \u2192 x \u21a6 x+1"},"headline":"A135838 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-a7ab0564dee15e53","kind":"fingerprint_related","node_a_id":"n-fe7b03c8bdb96043","node_b_id":"n-aa18ad1625904333","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-fe7b03c8bdb96043","other":{"id":"n-fe7b03c8bdb96043","name":"basis_map_A007318 of Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["10","5"],["60","60","15"]],"card_rows":[["1"],["10","5"],["60","60","15"],["280","420","210","35"]],"row_sums_preview":["1","15","135","945","5670","30618","153090","721710"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-79986ab5023c2764","steps":["basis A007318"],"edges":["e-484d449f4d0f99c5"],"text":"A178821 \u2192 basis A007318"},"headline":"A178821 \u2192 basis A007318","chain":null}},{"id":"l-a97f9252c1c59201","kind":"fingerprint_related","node_a_id":"n-aa18ad1625904333","node_b_id":"n-77621b0cfa8249dd","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-77621b0cfa8249dd","other":{"id":"n-77621b0cfa8249dd","name":"basis_map_A013611 of Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","16"],["40","160","160"]],"card_rows":[["1"],["8","16"],["40","160","160"],["160","960","1920","1280"]],"row_sums_preview":["1","24","360","4320","45360","435456","3919104","33592320"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a820c1859f68e71c","steps":["basis A013611"],"edges":["e-159425c642d412bb"],"text":"A178820 \u2192 basis A013611"},"headline":"A178820 \u2192 basis A013611","chain":null}},{"id":"l-aa931163c07532a4","kind":"fingerprint_related","node_a_id":"n-af416c91e2aef37c","node_b_id":"n-aa18ad1625904333","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-af416c91e2aef37c","other":{"id":"n-af416c91e2aef37c","name":"shift_x_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":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6","3"],["24","24","6"]],"card_rows":[["1"],["6","3"],["24","24","6"],["80","120","60","10"]],"row_sums_preview":["1","9","54","270","1215","5103","20412","78732"],"regions":["(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c8c19eba9d68881f","steps":["\u00d7(1+(3/2)n+(1/2)n\u00b2)","x \u21a6 x+1"],"edges":["e-739fcf83152eb2a2","e-34e3d9a08d46fb54"],"text":"A007318 \u2192 \u00d7(1+(3/2)n+(1/2)n\u00b2) \u2192 x \u21a6 x+1"},"headline":"A007318 \u2192 \u00d7(1+(3/2)n+(1/2)n\u00b2) \u2192 x \u21a6 x+1","chain":null}},{"id":"l-ae602d3dbce9b2bb","kind":"fingerprint_related","node_a_id":"n-890099793b4c816b","node_b_id":"n-aa18ad1625904333","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-890099793b4c816b","other":{"id":"n-890099793b4c816b","name":"mobius_x_over_x_plus_one of Riordan array (1-x,x/(1-x)).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["0","0","1"],["0","0","1","2"],["0","0","1","4","4"]],"card_rows":[["0","0","1"],["0","0","1","2"],["0","0","1","4","4"],["0","0","1","6","12","8"]],"row_sums_preview":["1","3","9","27","81","243","729","2187"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7717563017cb9e5f","steps":["x \u21a6 x/(x+1)"],"edges":["e-dd3ac95d772228bd"],"text":"A159854 \u2192 x \u21a6 x/(x+1)"},"headline":"A159854 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-b231abfdca24e181","kind":"fingerprint_related","node_a_id":"n-aa18ad1625904333","node_b_id":"n-9466c8441e24ce06","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-9466c8441e24ce06","other":{"id":"n-9466c8441e24ce06","name":"basis_map_A013609 of basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-1","-2"],["1","4","4"]],"card_rows":[["1"],["-1","-2"],["1","4","4"],["-1","-6","-12","-8"]],"row_sums_preview":["1","-3","9","-27","81","-243","729","-2187"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-477d9807e1cbb785","steps":["basis A013609"],"edges":["e-13dcb978bba71940"],"text":"basis-map coefficients of A030528, shifted \u2192 basis A013609"},"headline":"basis-map coefficients of A030528, shifted \u2192 basis A013609","chain":null}},{"id":"l-b34530bede9ad239","kind":"fingerprint_related","node_a_id":"n-aa18ad1625904333","node_b_id":"n-a76db9b78dbb5551","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-a76db9b78dbb5551","other":{"id":"n-a76db9b78dbb5551","name":"basis_map_A013611 of Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["12","24"],["84","336","336"]],"card_rows":[["1"],["12","24"],["84","336","336"],["448","2688","5376","3584"]],"row_sums_preview":["1","36","756","12096","163296","1959552","21555072","221709312"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-97228a88c81561dc","steps":["basis A013611"],"edges":["e-377b2cf400ddb478"],"text":"A178822 \u2192 basis A013611"},"headline":"A178822 \u2192 basis A013611","chain":null}},{"id":"l-ba6632924feca232","kind":"fingerprint_related","node_a_id":"n-af521840f49b6d03","node_b_id":"n-aa18ad1625904333","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-af521840f49b6d03","other":{"id":"n-af521840f49b6d03","name":"derivative 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":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["3"],["12","6"]],"card_rows":[["3"],["12","6"],["36","36","9"],["96","144","72","12"]],"row_sums_preview":["0","3","18","81","324","1215","4374","15309"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-51b4f5e7901c2b50","steps":["D"],"edges":["e-d19cf3982d12ec2e"],"text":"A120910 \u2192 D"},"headline":"A120910 \u2192 D","chain":null}},{"id":"l-bcdf05a8aa106fb8","kind":"fingerprint_related","node_a_id":"n-66b456586fffaa5f","node_b_id":"n-aa18ad1625904333","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-66b456586fffaa5f","other":{"id":"n-66b456586fffaa5f","name":"shift_x_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":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],[],["4","4","1"]],"card_rows":[["4","4","1"],["24","36","18","3"],["96","192","144","48","6"],["320","800","800","400","100","10"]],"row_sums_preview":["0","0","9","81","486","2430","10935","45927"],"regions":["(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c8c19eba9d68881f","steps":["\u00d7(\u2212(1/2)n+(1/2)n\u00b2)","x \u21a6 x+1"],"edges":["e-ccfad80d3bf960c1","e-7ca9df991456824b"],"text":"A007318 \u2192 \u00d7(\u2212(1/2)n+(1/2)n\u00b2) \u2192 x \u21a6 x+1"},"headline":"A007318 \u2192 \u00d7(\u2212(1/2)n+(1/2)n\u00b2) \u2192 x \u21a6 x+1","chain":null}},{"id":"l-bfe92a671914bd7b","kind":"fingerprint_related","node_a_id":"n-fff9dc1c6c2891b5","node_b_id":"n-aa18ad1625904333","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-fff9dc1c6c2891b5","other":{"id":"n-fff9dc1c6c2891b5","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*2^i.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","2"],["16","16","4"]],"card_rows":[["1"],["4","2"],["16","16","4"],["64","96","48","8"]],"row_sums_preview":["1","6","36","216","1296","7776","46656","279936"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d28ad9841fc73317","steps":["basis A007318"],"edges":["e-1756f066a00732e3"],"text":"A038208 \u2192 basis A007318"},"headline":"A038208 \u2192 basis A007318","chain":null}},{"id":"l-c9913511d9c41a48","kind":"fingerprint_related","node_a_id":"n-73bb59c333e132f0","node_b_id":"n-aa18ad1625904333","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-73bb59c333e132f0","other":{"id":"n-73bb59c333e132f0","name":"mobius_x_over_x_plus_one of basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-1","-2"],["1","4","4"]],"card_rows":[["1"],["-1","-2"],["1","4","4"],["-1","-6","-12","-8"]],"row_sums_preview":["1","-3","9","-27","81","-243","729","-2187"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-74c1e94669ff21ff","steps":["x \u21a6 x/(x+1)"],"edges":["e-26d899ee6632b86c"],"text":"basis-map coefficients of A113214, shifted \u2192 x \u21a6 x/(x+1)"},"headline":"basis-map coefficients of A113214, shifted \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-ce430f95b7677d38","kind":"fingerprint_related","node_a_id":"n-aa18ad1625904333","node_b_id":"n-4766ad62f3b3a125","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-4766ad62f3b3a125","other":{"id":"n-4766ad62f3b3a125","name":"basis_map_A013611 of Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","8"],["24","96","96"]],"card_rows":[["1"],["4","8"],["24","96","96"],["192","1152","2304","1536"]],"row_sums_preview":["1","12","216","5184","155520","5598720","235146240","11287019520"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-75a6475b3e851dc5","steps":["basis A013611"],"edges":["e-fb2d67190f0fa658"],"text":"A156992 \u2192 basis A013611"},"headline":"A156992 \u2192 basis A013611","chain":null}},{"id":"l-ce92feaef53f50c9","kind":"fingerprint_related","node_a_id":"n-aa18ad1625904333","node_b_id":"n-57f251ac09effa9f","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-57f251ac09effa9f","other":{"id":"n-57f251ac09effa9f","name":"basis_map_A013611 of Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["10","20"],["60","240","240"]],"card_rows":[["1"],["10","20"],["60","240","240"],["280","1680","3360","2240"]],"row_sums_preview":["1","30","540","7560","90720","979776","9797760","92378880"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-79986ab5023c2764","steps":["basis A013611"],"edges":["e-75b8144fcd4b7728"],"text":"A178821 \u2192 basis A013611"},"headline":"A178821 \u2192 basis A013611","chain":null}},{"id":"l-d01d84ec8275c0b0","kind":"fingerprint_related","node_a_id":"n-aa18ad1625904333","node_b_id":"n-fdb3be84f583ed95","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-fdb3be84f583ed95","other":{"id":"n-fdb3be84f583ed95","name":"basis_map_A013609 of basis-map coefficients of A scaled Hermite triangle.","first_index":1,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1","4","4"],["1","6","12","8"],["1","8","24","32","16"]],"card_rows":[["1","4","4"],["1","6","12","8"],["1","8","24","32","16"],["1","10","40","80","80","32"]],"row_sums_preview":["9","27","81","243","729","2187","6561","19683"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5fe240122e98ffb3","steps":["basis A013609"],"edges":["e-7bee75fc05394bb8"],"text":"basis-map coefficients of A112227 \u2192 basis A013609"},"headline":"basis-map coefficients of A112227 \u2192 basis A013609","chain":null}},{"id":"l-d1a9475af341084a","kind":"fingerprint_related","node_a_id":"n-ec536fa823a7f7aa","node_b_id":"n-aa18ad1625904333","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-ec536fa823a7f7aa","other":{"id":"n-ec536fa823a7f7aa","name":"derivative of Triangle of coefficients in expansion of (1+2*x)^n.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","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","24"],["8","48","96","64"]],"row_sums_preview":["0","2","12","54","216","810","2916","10206"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-71965e79e47a4526","steps":["D"],"edges":["e-9f3f778aa1812acb"],"text":"A013609 \u2192 D"},"headline":"A013609 \u2192 D","chain":null}},{"id":"l-e3f7a244e20fe63d","kind":"fingerprint_related","node_a_id":"n-aa18ad1625904333","node_b_id":"n-0793a8fd2b196f0d","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-0793a8fd2b196f0d","other":{"id":"n-0793a8fd2b196f0d","name":"basis_map_A013609 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":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":1,"preview_rows":[["1"],["-1","-2"],["1","4","4"]],"card_rows":[["1"],["-1","-2"],["1","4","4"],["-1","-6","-12","-8"]],"row_sums_preview":["1","-3","9","-27","81","-243","729","-2187"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3cc7875cf81a5a84","steps":["basis A013609"],"edges":["e-66f8525b021adc45"],"text":"basis-map coefficients of A026729, shifted \u2192 basis A013609"},"headline":"basis-map coefficients of A026729, shifted \u2192 basis A013609","chain":null}},{"id":"l-e6fd2714a2706eaa","kind":"fingerprint_related","node_a_id":"n-aa18ad1625904333","node_b_id":"n-feee915c2922c082","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-feee915c2922c082","other":{"id":"n-feee915c2922c082","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*3^j.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["6","12"],["36","144","144"]],"card_rows":[["1"],["6","12"],["36","144","144"],["216","1296","2592","1728"]],"row_sums_preview":["1","18","324","5832","104976","1889568","34012224","612220032"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8d88ef9de74b305f","steps":["basis A013611"],"edges":["e-fc7ebcb9132c171a"],"text":"A038221 \u2192 basis A013611"},"headline":"A038221 \u2192 basis A013611","chain":null}},{"id":"l-ef227bdd2d2799a4","kind":"fingerprint_related","node_a_id":"n-aa18ad1625904333","node_b_id":"n-fbc103865e18ab22","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-fbc103865e18ab22","other":{"id":"n-fbc103865e18ab22","name":"mobius_x_over_x_plus_one of Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["5","10"],["15","60","60"]],"card_rows":[["1"],["5","10"],["15","60","60"],["35","210","420","280"]],"row_sums_preview":["1","15","135","945","5670","30618","153090","721710"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-79986ab5023c2764","steps":["x \u21a6 x/(x+1)"],"edges":["e-da74bc8fbb9fce7e"],"text":"A178821 \u2192 x \u21a6 x/(x+1)"},"headline":"A178821 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-f0588e4b9c0e6d87","kind":"fingerprint_related","node_a_id":"n-aa18ad1625904333","node_b_id":"n-11dd292935cd7dbf","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-11dd292935cd7dbf","other":{"id":"n-11dd292935cd7dbf","name":"mobius_x_over_x_plus_one of Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","8"],["10","40","40"]],"card_rows":[["1"],["4","8"],["10","40","40"],["20","120","240","160"]],"row_sums_preview":["1","12","90","540","2835","13608","61236","262440"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a820c1859f68e71c","steps":["x \u21a6 x/(x+1)"],"edges":["e-384d0b9c26a615c7"],"text":"A178820 \u2192 x \u21a6 x/(x+1)"},"headline":"A178820 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-f2d6de002057f782","kind":"fingerprint_related","node_a_id":"n-aa18ad1625904333","node_b_id":"n-4dab44b954732fbb","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-4dab44b954732fbb","other":{"id":"n-4dab44b954732fbb","name":"basis_map_A013611 of Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","8"],["24","96","96"]],"card_rows":[["1"],["4","8"],["24","96","96"],["160","960","1920","1280"]],"row_sums_preview":["1","12","216","4320","90720","1959552","43110144","960740352"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-dd77bf712b1df996","steps":["basis A013611"],"edges":["e-6d76bdb8087cd7e0"],"text":"A356546 \u2192 basis A013611"},"headline":"A356546 \u2192 basis A013611","chain":null}},{"id":"l-f4327572b7c90d3f","kind":"fingerprint_related","node_a_id":"n-aa18ad1625904333","node_b_id":"n-c201d29a62400440","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-c201d29a62400440","other":{"id":"n-c201d29a62400440","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["576","2304","2304"],["13824","82944","165888","110592"],["331776","2654208","7962624","10616832","5308416"]],"card_rows":[["576","2304","2304"],["13824","82944","165888","110592"],["331776","2654208","7962624","10616832","5308416"],["7962624","79626240","318504960","637009920","637009920","254803968"]],"row_sums_preview":["5184","373248","26873856","1934917632","139314069504","10030613004288","722204136308736","51998697814228992"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c1532953a6736702","steps":["basis A013611"],"edges":["e-c462762e7768ae01"],"text":"A038338 \u2192 basis A013611"},"headline":"A038338 \u2192 basis A013611","chain":null}},{"id":"l-f5f2a38c7f9f6f2a","kind":"fingerprint_related","node_a_id":"n-aa18ad1625904333","node_b_id":"n-bc7333ad044cbf73","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-bc7333ad044cbf73","other":{"id":"n-bc7333ad044cbf73","name":"basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j).","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["8","4"],["64","64","16"]],"card_rows":[["1"],["8","4"],["64","64","16"],["512","768","384","64"]],"row_sums_preview":["1","12","144","1728","20736","248832","2985984","35831808"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f211cb6153c37c6d","steps":["basis A013611"],"edges":["e-267942f18014ef6e"],"text":"A027466 \u2192 basis A013611"},"headline":"A027466 \u2192 basis A013611","chain":null}},{"id":"l-fbef366e34511eba","kind":"fingerprint_related","node_a_id":"n-aa18ad1625904333","node_b_id":"n-de403801a492a76b","fingerprint":"f6-11aec07ce60b6d61","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-11aec07ce60b6d61"},"other_id":"n-de403801a492a76b","other":{"id":"n-de403801a492a76b","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.","first_index":0,"fingerprint":"f6-11aec07ce60b6d61","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["256","256","64"],["4096","6144","3072","512"],["65536","131072","98304","32768","4096"]],"card_rows":[["256","256","64"],["4096","6144","3072","512"],["65536","131072","98304","32768","4096"],["1048576","2621440","2621440","1310720","327680","32768"]],"row_sums_preview":["576","13824","331776","7962624","191102976","4586471424","110075314176","2641807540224"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a93042cd70c4761e","steps":["basis A013609"],"edges":["e-6efc8fa5932928ce"],"text":"A038330 \u2192 basis A013609"},"headline":"A038330 \u2192 basis A013609","chain":null}},{"id":"l-0ad7a41dbe672dc7","kind":"oeis_match","node_a_id":"n-aa18ad1625904333","node_b_id":null,"fingerprint":null,"oeis_id":"A069996","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A069996","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A069996","relation":"match_row_sums"}],"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,    8;\n  2:   9,   36,    36;\n  3:  16,   96,   192,   128;\n  4:  25,  200,   600,   800,    400;\n  5:  36,  360,  1440,  2880,   2880,   1152;\n  6:  49,  588,  2940,  7840,  11760,   9408,   3136;\n  7:  64,  896,  5376, 17920,  35840,  43008,  28672,   8192;\n  8:  81, 1296,  9072, 36288,  90720, 145152, 145152,  82944,  20736;\n  9: 100, 1800, 14400, 67200, 201600, 403200, 537600, 460800, 230400, 51200;\n","recurrence_sentence":"The row polynomials satisfy (n^2+n^2*x)*P(n,x) = (n^2+2*n+1+(3*n^2+6*n+3)*x+(2*n^2+4*n+2)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((n^2 + 2*n + 1 + (3*n^2 + 6*n + 3)*x + (2*n^2 + 4*n + 2)*x^2)*P[n - 1, x])/(n^2 + n^2*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((n^2 + 2*n + 1 + (3*n^2 + 6*n + 3)*x + (2*n^2 + 4*n + 2)*x^2)*v[i - 1])/(n^2 + n^2*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-c8c19eba9d68881f","steps":["\u00d7(1+2n+n\u00b2)","x \u21a6 x/(x+1)"],"edges":["e-b78deffc64bc1942","e-963b3afafd9f7f70"],"text":"A007318 \u2192 \u00d7(1+2n+n\u00b2) \u2192 x \u21a6 x/(x+1)"},"headline":"A007318 \u2192 \u00d7(1+2n+n\u00b2) \u2192 x \u21a6 x/(x+1)","chain":null}
