{"id":"n-c990ad92ffd076e4","revision":1,"first_index":1,"fingerprint":"f6-b5f1389370a379f1","name":"reversal of Triangle read by rows: 2-Stirling numbers of the second kind.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1","2"],["1","5","4"],["1","9","19","8"],["1","14","55","65","16"],["1","20","125","285","211","32"],["1","27","245","910","1351","665","64"],["1","35","434","2380","5901","6069","2059","128"],["1","44","714","5418","20181","35574","26335","6305","256"],["1","54","1110","11130","58107","156660","204205","111645","19171","512"],["1","65","1650","21120","147147","563409","1144165","1132670","465751","58025","1024"],["1","77","2365","37620","337227","1740585","5088028","7997660","6129101","1921029","175099","2048"],["1","90","3289","63635","713427","4775628","19012708","43613856","54115061","32566534","7859215","527345","4096"],["1","104","4459","103103","1413412","11909898","61993360","195715520","359412053","357256900","170691885","31964205","1586131","8192"],["1","119","5915","161070","2650648","27457430","181092340","753655760","1925136213","2873141271","2314233285","885423630","129442951","4766585","16384"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","10","37","151","674","3263","17007","94828","562595","3535027","23430840","163254885","1192059223","9097183602"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","0"],["0","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":165072,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-e104c7d9553deadb"]},{"id":165073,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-e104c7d9553deadb"]},{"id":165074,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-e104c7d9553deadb"]}],"region_exact":{},"incoming_edges":[{"id":"e-e104c7d9553deadb","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":1,\"first_degree\":0,\"step\":1}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-c990ad92ffd076e4","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":"reversed","lean_declarations":["RealRooted.X_pow_mul_reverse_eq_zero_or_splits","RealRooted.reflect_eq_zero_or_splits"],"certificate_hash":"594bb4d3fa44844f2cfa465f1ec030b2bbe6493a24939715f6fb23b55c7e24fb","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-6aaec68d653cbdb0","role":"sequence","name":"Triangle read by rows: 2-Stirling numbers of the second kind.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-0bd644242fa7fcfb","kind":"fingerprint_related","node_a_id":"n-c8044fd3d7ed3bae","node_b_id":"n-c990ad92ffd076e4","fingerprint":"f6-b5f1389370a379f1","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b5f1389370a379f1"},"other_id":"n-c8044fd3d7ed3bae","other":{"id":"n-c8044fd3d7ed3bae","name":"basis_map_A008277 of Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-b5f1389370a379f1","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["10","5"],["60","75","15"]],"card_rows":[["1"],["10","5"],["60","75","15"],["280","665","315","35"]],"row_sums_preview":["1","15","150","1295","10570","84924","685230","5612310"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-79986ab5023c2764","steps":["basis A008277"],"edges":["e-2482ec2370e1f9d0"],"text":"A178821 \u2192 basis A008277"},"headline":"A178821 \u2192 basis A008277","chain":null}},{"id":"l-0ed286e42aeede30","kind":"fingerprint_related","node_a_id":"n-c990ad92ffd076e4","node_b_id":"n-11f206dfc8a798d6","fingerprint":"f6-b5f1389370a379f1","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b5f1389370a379f1"},"other_id":"n-11f206dfc8a798d6","other":{"id":"n-11f206dfc8a798d6","name":"basis_map_A008277 of Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(n).","first_index":0,"fingerprint":"f6-b5f1389370a379f1","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","1"],["12","15","3"]],"card_rows":[["1"],["2","1"],["12","15","3"],["120","285","135","15"]],"row_sums_preview":["1","3","30","555","15855","636930","33918885","2298240945"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c7809537965f400d","steps":["basis A008277"],"edges":["e-380a606cb73e0af7"],"text":"A085881 \u2192 basis A008277"},"headline":"A085881 \u2192 basis A008277","chain":null}},{"id":"l-2bf80b52d77105c7","kind":"fingerprint_related","node_a_id":"n-34783469fb5242d8","node_b_id":"n-c990ad92ffd076e4","fingerprint":"f6-b5f1389370a379f1","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b5f1389370a379f1"},"other_id":"n-34783469fb5242d8","other":{"id":"n-34783469fb5242d8","name":"basis_map_A008277 of Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-b5f1389370a379f1","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["8","4"],["40","50","10"]],"card_rows":[["1"],["8","4"],["40","50","10"],["160","380","180","20"]],"row_sums_preview":["1","12","100","740","5285","37744","274092","2040840"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a820c1859f68e71c","steps":["basis A008277"],"edges":["e-96de39e5a8006173"],"text":"A178820 \u2192 basis A008277"},"headline":"A178820 \u2192 basis A008277","chain":null}},{"id":"l-3dc7fcb5d9a2280e","kind":"fingerprint_related","node_a_id":"n-c514ba2a1328f4f7","node_b_id":"n-c990ad92ffd076e4","fingerprint":"f6-b5f1389370a379f1","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b5f1389370a379f1"},"other_id":"n-c514ba2a1328f4f7","other":{"id":"n-c514ba2a1328f4f7","name":"basis_map_A008277 of binomial","first_index":0,"fingerprint":"f6-b5f1389370a379f1","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","1"],["4","5","1"]],"card_rows":[["1"],["2","1"],["4","5","1"],["8","19","9","1"]],"row_sums_preview":["1","3","10","37","151","674","3263","17007"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-0990b5be9f344750","steps":["reverse"],"edges":["e-a499c43746a8b06c"],"text":"A126351 \u2192 reverse"},"headline":"A126351 \u2192 reverse","chain":null}},{"id":"l-3ff7d3197acbcbdc","kind":"fingerprint_related","node_a_id":"n-272049576da882c4","node_b_id":"n-c990ad92ffd076e4","fingerprint":"f6-b5f1389370a379f1","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b5f1389370a379f1"},"other_id":"n-272049576da882c4","other":{"id":"n-272049576da882c4","name":"basis_map_A008277 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-b5f1389370a379f1","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["3"],["30","15"],["180","225","45"]],"card_rows":[["3"],["30","15"],["180","225","45"],["840","1995","945","105"]],"row_sums_preview":["3","45","450","3885","31710","254772","2055690","16836930"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-dd4aa31f3d11d152","steps":["basis A008277"],"edges":["e-19677b98e028e07a"],"text":"A344565 \u2192 basis A008277"},"headline":"A344565 \u2192 basis A008277","chain":null}},{"id":"l-403ad5acd0761959","kind":"fingerprint_related","node_a_id":"n-6aaec68d653cbdb0","node_b_id":"n-c990ad92ffd076e4","fingerprint":"f6-b5f1389370a379f1","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b5f1389370a379f1"},"other_id":"n-6aaec68d653cbdb0","other":{"id":"n-6aaec68d653cbdb0","name":"Triangle read by rows: 2-Stirling numbers of the second kind.","first_index":1,"fingerprint":"f6-b5f1389370a379f1","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","1"],["4","5","1"]],"card_rows":[["1"],["2","1"],["4","5","1"],["8","19","9","1"]],"row_sums_preview":["1","3","10","37","151","674","3263","17007"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":["A143494"],"derivation":{"root":"n-6aaec68d653cbdb0","steps":[],"edges":[],"text":"A143494"},"headline":"A143494: Triangle read by rows: 2-Stirling numbers of the second kind.","chain":null}},{"id":"l-4b4b3f55ef1f8083","kind":"fingerprint_related","node_a_id":"n-c072d889dfa1d499","node_b_id":"n-c990ad92ffd076e4","fingerprint":"f6-b5f1389370a379f1","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b5f1389370a379f1"},"other_id":"n-c072d889dfa1d499","other":{"id":"n-c072d889dfa1d499","name":"basis_map_A008277 of Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-b5f1389370a379f1","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["12","6"],["84","105","21"]],"card_rows":[["1"],["12","6"],["84","105","21"],["448","1064","504","56"]],"row_sums_preview":["1","18","210","2072","19026","169848","1507506","13469544"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-97228a88c81561dc","steps":["basis A008277"],"edges":["e-af56d17dbcb6f76e"],"text":"A178822 \u2192 basis A008277"},"headline":"A178822 \u2192 basis A008277","chain":null}},{"id":"l-60adc48b445188c3","kind":"fingerprint_related","node_a_id":"n-29f388dd4604fb27","node_b_id":"n-c990ad92ffd076e4","fingerprint":"f6-b5f1389370a379f1","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b5f1389370a379f1"},"other_id":"n-29f388dd4604fb27","other":{"id":"n-29f388dd4604fb27","name":"basis_map_A008277 of Triangle read by rows: T(n,k) = 2^floor(n/2)*binomial(n-1,k-1).","first_index":0,"fingerprint":"f6-b5f1389370a379f1","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4","2"],["8","10","2"]],"card_rows":[["1"],["4","2"],["8","10","2"],["32","76","36","4"]],"row_sums_preview":["1","6","20","148","604","5392","26104","272112"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a624e5aba8e1c39e","steps":["basis A008277"],"edges":["e-f15543b521e0834d"],"text":"A135838 \u2192 basis A008277"},"headline":"A135838 \u2192 basis A008277","chain":null}},{"id":"l-8b3c718b8683e9e2","kind":"fingerprint_related","node_a_id":"n-0990b5be9f344750","node_b_id":"n-c990ad92ffd076e4","fingerprint":"f6-b5f1389370a379f1","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b5f1389370a379f1"},"other_id":"n-0990b5be9f344750","other":{"id":"n-0990b5be9f344750","name":"Triangle read by rows: matrix product of the Stirling numbers of the second kind with the binomial coefficients.","first_index":0,"fingerprint":"f6-b5f1389370a379f1","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","2"],["1","5","4"]],"card_rows":[["1"],["1","2"],["1","5","4"],["1","9","19","8"]],"row_sums_preview":["1","3","10","37","151","674","3263","17007"],"regions":["(-inf, 0)"],"region_exact":{},"oeis_ids":["A126351"],"derivation":{"root":"n-0990b5be9f344750","steps":[],"edges":[],"text":"A126351"},"headline":"A126351: Triangle read by rows: matrix product of the Stirling numbers of the second kind with the binomial coefficients.","chain":null}},{"id":"l-9a6c62f689291a03","kind":"fingerprint_related","node_a_id":"n-d7033d5397b0761f","node_b_id":"n-c990ad92ffd076e4","fingerprint":"f6-b5f1389370a379f1","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b5f1389370a379f1"},"other_id":"n-d7033d5397b0761f","other":{"id":"n-d7033d5397b0761f","name":"basis_map_A008277 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-b5f1389370a379f1","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","1"],["8","10","2"]],"card_rows":[["1"],["2","1"],["8","10","2"],["40","95","45","5"]],"row_sums_preview":["1","3","20","185","2114","28308","430716","7296003"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a98fae1bf9eeb913","steps":["basis A008277"],"edges":["e-63a786f336006cd7"],"text":"A085880 \u2192 basis A008277"},"headline":"A085880 \u2192 basis A008277","chain":null}},{"id":"l-a78625808277e0a0","kind":"fingerprint_related","node_a_id":"n-873e284fbdbad5b5","node_b_id":"n-c990ad92ffd076e4","fingerprint":"f6-b5f1389370a379f1","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b5f1389370a379f1"},"other_id":"n-873e284fbdbad5b5","other":{"id":"n-873e284fbdbad5b5","name":"basis_map_A008277 of Triangle whose (i,j)-th entry is binomial(i,j)*2^i.","first_index":0,"fingerprint":"f6-b5f1389370a379f1","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4","2"],["16","20","4"]],"card_rows":[["1"],["4","2"],["16","20","4"],["64","152","72","8"]],"row_sums_preview":["1","6","40","296","2416","21568","208832","2176896"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d28ad9841fc73317","steps":["basis A008277"],"edges":["e-21aff6157efb6757"],"text":"A038208 \u2192 basis A008277"},"headline":"A038208 \u2192 basis A008277","chain":null}},{"id":"l-aac5a4a648d59e7c","kind":"fingerprint_related","node_a_id":"n-a514cae29d6b9cc2","node_b_id":"n-c990ad92ffd076e4","fingerprint":"f6-b5f1389370a379f1","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b5f1389370a379f1"},"other_id":"n-a514cae29d6b9cc2","other":{"id":"n-a514cae29d6b9cc2","name":"basis_map_A008277 of basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-b5f1389370a379f1","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":0,"preview_rows":[["1"],["-2","-1"],["4","5","1"]],"card_rows":[["1"],["-2","-1"],["4","5","1"],["-8","-19","-9","-1"]],"row_sums_preview":["1","-3","10","-37","151","-674","3263","-17007"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-74c1e94669ff21ff","steps":["basis A008277"],"edges":["e-0bd7ea1430928905"],"text":"basis-map coefficients of A113214, shifted \u2192 basis A008277"},"headline":"basis-map coefficients of A113214, shifted \u2192 basis A008277","chain":null}},{"id":"l-b62b0c96aafe0ed3","kind":"fingerprint_related","node_a_id":"n-c990ad92ffd076e4","node_b_id":"n-b1fe43ecd25295ad","fingerprint":"f6-b5f1389370a379f1","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b5f1389370a379f1"},"other_id":"n-b1fe43ecd25295ad","other":{"id":"n-b1fe43ecd25295ad","name":"basis_map_A008277 of Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).","first_index":0,"fingerprint":"f6-b5f1389370a379f1","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4","2"],["24","30","6"]],"card_rows":[["1"],["4","2"],["24","30","6"],["160","380","180","20"]],"row_sums_preview":["1","6","60","740","10570","169848","3015012","58368024"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-dd77bf712b1df996","steps":["basis A008277"],"edges":["e-398f569c764d9a1d"],"text":"A356546 \u2192 basis A008277"},"headline":"A356546 \u2192 basis A008277","chain":null}},{"id":"l-c2034eed2b7401ef","kind":"fingerprint_related","node_a_id":"n-c990ad92ffd076e4","node_b_id":"n-646faea170766425","fingerprint":"f6-b5f1389370a379f1","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b5f1389370a379f1"},"other_id":"n-646faea170766425","other":{"id":"n-646faea170766425","name":"basis_map_A008277 of Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*11^j.","first_index":0,"fingerprint":"f6-b5f1389370a379f1","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["22","11"],["484","605","121"]],"card_rows":[["1"],["22","11"],["484","605","121"],["10648","25289","11979","1331"]],"row_sums_preview":["1","33","1210","49247","2210791","108548374","5780603543","331418317197"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8064e72cf80883a7","steps":["basis A008277"],"edges":["e-f221360d3c3c712f"],"text":"A038325 \u2192 basis A008277"},"headline":"A038325 \u2192 basis A008277","chain":null}},{"id":"l-c2979350d7bfe7e0","kind":"fingerprint_related","node_a_id":"n-005d2be3e1674578","node_b_id":"n-c990ad92ffd076e4","fingerprint":"f6-b5f1389370a379f1","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b5f1389370a379f1"},"other_id":"n-005d2be3e1674578","other":{"id":"n-005d2be3e1674578","name":"basis_map_A008277 of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).","first_index":0,"fingerprint":"f6-b5f1389370a379f1","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["2"],["4","2"],["8","10","2"]],"card_rows":[["2"],["4","2"],["8","10","2"],["16","38","18","2"]],"row_sums_preview":["2","6","20","74","302","1348","6526","34014"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-089a9b4f824a1c20","steps":["basis A008277"],"edges":["e-fac469530424e761"],"text":"A028326 \u2192 basis A008277"},"headline":"A028326 \u2192 basis A008277","chain":null}},{"id":"l-c7e6931ab41c926a","kind":"fingerprint_related","node_a_id":"n-c990ad92ffd076e4","node_b_id":"n-d06008b4ada4e713","fingerprint":"f6-b5f1389370a379f1","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b5f1389370a379f1"},"other_id":"n-d06008b4ada4e713","other":{"id":"n-d06008b4ada4e713","name":"basis_map_A008277 of Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).","first_index":0,"fingerprint":"f6-b5f1389370a379f1","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","1"],["8","10","2"]],"card_rows":[["1"],["2","1"],["8","10","2"],["48","114","54","6"]],"row_sums_preview":["1","3","20","222","3624","80880","2349360","85715280"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cb5d49e71e46f2ed","steps":["basis A008277"],"edges":["e-73c89ff38d5bbc45"],"text":"A196347 \u2192 basis A008277"},"headline":"A196347 \u2192 basis A008277","chain":null}},{"id":"l-c9b6beccf9edaa9d","kind":"fingerprint_related","node_a_id":"n-c990ad92ffd076e4","node_b_id":"n-085c06a40df81890","fingerprint":"f6-b5f1389370a379f1","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b5f1389370a379f1"},"other_id":"n-085c06a40df81890","other":{"id":"n-085c06a40df81890","name":"basis_map_A008277 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.","first_index":0,"fingerprint":"f6-b5f1389370a379f1","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["576","720","144"],["13824","32832","15552","1728"],["331776","1347840","1140480","290304","20736"]],"card_rows":[["576","720","144"],["13824","32832","15552","1728"],["331776","1347840","1140480","290304","20736"],["7962624","52503552","70917120","31104000","4976640","248832"]],"row_sums_preview":["1440","63936","3131136","167712768","9743265792","609391558656","40774304268288","2902866627133440"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c1532953a6736702","steps":["basis A008277"],"edges":["e-44f842c6e5783c9b"],"text":"A038338 \u2192 basis A008277"},"headline":"A038338 \u2192 basis A008277","chain":null}},{"id":"l-d0046eff7210ca08","kind":"fingerprint_related","node_a_id":"n-c990ad92ffd076e4","node_b_id":"n-50766ce7e49984e9","fingerprint":"f6-b5f1389370a379f1","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b5f1389370a379f1"},"other_id":"n-50766ce7e49984e9","other":{"id":"n-50766ce7e49984e9","name":"basis_map_A008277 of Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.","first_index":0,"fingerprint":"f6-b5f1389370a379f1","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4","2"],["24","30","6"]],"card_rows":[["1"],["4","2"],["24","30","6"],["192","456","216","24"]],"row_sums_preview":["1","6","60","888","18120","485280","16445520","685722240"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-75a6475b3e851dc5","steps":["basis A008277"],"edges":["e-d06f83defd222c3e"],"text":"A156992 \u2192 basis A008277"},"headline":"A156992 \u2192 basis A008277","chain":null}},{"id":"l-ebffc03806a86180","kind":"fingerprint_related","node_a_id":"n-c990ad92ffd076e4","node_b_id":"n-9b47026307c031f7","fingerprint":"f6-b5f1389370a379f1","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b5f1389370a379f1"},"other_id":"n-9b47026307c031f7","other":{"id":"n-9b47026307c031f7","name":"basis_map_A008277 of Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*3^j.","first_index":0,"fingerprint":"f6-b5f1389370a379f1","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["6","3"],["36","45","9"]],"card_rows":[["1"],["6","3"],["36","45","9"],["216","513","243","27"]],"row_sums_preview":["1","9","90","999","12231","163782","2378727","37194309"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8d88ef9de74b305f","steps":["basis A008277"],"edges":["e-d86682f478f8bbfa"],"text":"A038221 \u2192 basis A008277"},"headline":"A038221 \u2192 basis A008277","chain":null}},{"id":"l-00c5f633111b7378","kind":"oeis_match","node_a_id":"n-c990ad92ffd076e4","node_b_id":null,"fingerprint":null,"oeis_id":"A005493","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A005493","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-3a114305c1ccd844","kind":"oeis_match","node_a_id":"n-c990ad92ffd076e4","node_b_id":null,"fingerprint":null,"oeis_id":"A126351","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":10,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":10,"target":"triangle","terms_compared":55},"in_catalog":true,"oeis_id":"A126351","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-87e11d27ffe40d4a","kind":"oeis_match","node_a_id":"n-c990ad92ffd076e4","node_b_id":null,"fingerprint":null,"oeis_id":"A138378","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A138378","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-abb130cda7dfad81","kind":"oeis_match","node_a_id":"n-c990ad92ffd076e4","node_b_id":null,"fingerprint":null,"oeis_id":"A143494","in_catalog":1,"target":"reversed_triangle","row_alignment":"tabl","rows_matched":10,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":2,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":10,"target":"reversed_triangle","terms_compared":55},"in_catalog":true,"oeis_id":"A143494","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A143494","relation":"match_reversed_triangle"},{"oeis_id":"A005493","relation":"match_row_sums"},{"oeis_id":"A138378","relation":"match_row_sums"},{"oeis_id":"A126351","relation":"match_triangle"}],"bfiles":[{"node_id":"n-c990ad92ffd076e4","oeis_id":"A005493","target":"row_sums","oeis_offset":0,"first_row":1,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":24,"oeis_terms_available":24,"term_count":200,"raw_size":25816},{"node_id":"n-c990ad92ffd076e4","oeis_id":"A126351","target":"triangle","oeis_offset":1,"first_row":1,"rows_used":44,"terms_matched":55,"oeis_terms_agreeing":55,"oeis_terms_available":55,"term_count":990,"raw_size":22434},{"node_id":"n-c990ad92ffd076e4","oeis_id":"A138378","target":"row_sums","oeis_offset":1,"first_row":1,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":24,"oeis_terms_available":24,"term_count":200,"raw_size":25818},{"node_id":"n-c990ad92ffd076e4","oeis_id":"A143494","target":"reversed_triangle","oeis_offset":2,"first_row":1,"rows_used":44,"terms_matched":55,"oeis_terms_agreeing":55,"oeis_terms_available":55,"term_count":990,"raw_size":22436}],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 1,  2;\n   3: 1,  5,    4;\n   4: 1,  9,   19,     8;\n   5: 1, 14,   55,    65,    16;\n   6: 1, 20,  125,   285,   211,     32;\n   7: 1, 27,  245,   910,  1351,    665,     64;\n   8: 1, 35,  434,  2380,  5901,   6069,   2059,    128;\n   9: 1, 44,  714,  5418, 20181,  35574,  26335,   6305,   256;\n  10: 1, 54, 1110, 11130, 58107, 156660, 204205, 111645, 19171, 512;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+n*x)*P(n-1,x) - x^2*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(1 + n*x)*P[n - 1, x] - x^2*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (1 + n*x)*v[i - 1] - x^2*deriv(v[i - 1], x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}},{"oeis_id":"A126351","target":"triangle","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 1,  2;\n   3: 1,  5,    4;\n   4: 1,  9,   19,     8;\n   5: 1, 14,   55,    65,    16;\n   6: 1, 20,  125,   285,   211,     32;\n   7: 1, 27,  245,   910,  1351,    665,     64;\n   8: 1, 35,  434,  2380,  5901,   6069,   2059,    128;\n   9: 1, 44,  714,  5418, 20181,  35574,  26335,   6305,   256;\n  10: 1, 54, 1110, 11130, 58107, 156660, 204205, 111645, 19171, 512;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+n*x)*P(n-1,x) - x^2*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(1 + n*x)*P[n - 1, x] - x^2*D[P[n - 1, x], x]];\nTable[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (1 + n*x)*v[i - 1] - x^2*deriv(v[i - 1], x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":true}},{"oeis_id":"A143494","target":"reversed_triangle","first_row":1,"oeis_offset":2,"reversed":true,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   2:   1;\n   3:   2,     1;\n   4:   4,     5,      1;\n   5:   8,    19,      9,      1;\n   6:  16,    65,     55,     14,      1;\n   7:  32,   211,    285,    125,     20,     1;\n   8:  64,   665,   1351,    910,    245,    27,     1;\n   9: 128,  2059,   6069,   5901,   2380,   434,    35,    1;\n  10: 256,  6305,  26335,  35574,  20181,  5418,   714,   44,  1;\n  11: 512, 19171, 111645, 204205, 156660, 58107, 11130, 1110, 54, 1;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+(n-1)*x)*P(n-1,x) - x^2*d/dx P(n-1,x) for n >= 3, with P(2,x) = 1.","mathematica":"P[2, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(1 + (n - 1)*x)*P[n - 1, x] - x^2*D[P[n - 1, x], x]];\nTable[Reverse[PadRight[CoefficientList[P[n, x], x], n - 1]], {n, 2, 11}] // Flatten","pari":"P(N) = {my(v = vector(N - 1)); v[1] = 1; for(n = 3, N, my(i = n - 1); v[i] = (1 + (n - 1)*x)*v[i - 1] - x^2*deriv(v[i - 1], x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - 1 - #c)); c = Vecrev(c); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i + 1, v[i])))};\nT(11)","reading":{"first_row":1,"offset":2,"reversed":true,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-6aaec68d653cbdb0","steps":["reverse"],"edges":["e-e104c7d9553deadb"],"text":"A143494 \u2192 reverse"},"headline":"A143494 \u2192 reverse","chain":null}
