{"id":"n-fefc87d068bda822","revision":1,"first_index":0,"fingerprint":"f6-53af44bcc1c9acce","name":"Triangle read by rows, Stirling cycle numbers of order 3, T(n,n) = 1, T(n,k) = 0 if k<0 or k>n, otherwise T(n,k) = T(n-1,k-1)+(n-1)^3*T(n-1,k), for n>=0 and 0<=k<=n.","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"],["0","1"],["0","1","1"],["0","8","9","1"],["0","216","251","36","1"],["0","13824","16280","2555","100","1"],["0","1728000","2048824","335655","15055","225","1"],["0","373248000","444273984","74550304","3587535","63655","441","1"],["0","128024064000","152759224512","26015028256","1305074809","25421200","214918","784","1"],["0","65548320768000","78340747014144","13472453691584","694213330464","14320729209","135459216","616326","1296","1"],["0","47784725839872000","57175952894078976","9899759488178880","519553971599840","11134024923825","113070497673","584760870","1561110","2025","1"],["0","47784725839872000000","57223737619918848000","9956935441072958976","529453731088018880","11653578895424840","124204522596825","697831367673","2145870870","3586110","3025","1"],["0","63601470092869632000000","76212579497951858688000","13309904809688027245056","714659851519226088256","16040367240898480920","176969798471798915","1053018072969588","3553985495643","6918983280","7612385","4356","1"],["0","109903340320478724096000000","131758938842553681444864000","23075728090638862938144768","1248242128234910707751424","28432414443791801118016","321844179000167006040","1996585028563246979","7194305009440692","15509988603483","20073184560","15139553","6084","1"],["0","241457638684091756838912000000","289584291977410916858462208000","50829133553976135556548919296","2765463683822737687868023296","63714256661245497764032576","735524075707158713387896","4708341486753620618903","17802473134304447303","41269749971292843","59610775081803","53334782501","28506101","8281","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","1","2","18","504","32760","4127760","895723920","308129028480","158070191610240","115391239875475200","115506631115350675200","153854832645647099366400","266015005644323834804505600","584700982406223788900303308800"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A269947","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1","1"],["3","0"],["-3","0"],["1","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":159827,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-f7365f1dbad216ac"]},{"id":159828,"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-f7365f1dbad216ac"]},{"id":159829,"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-f7365f1dbad216ac"]}],"region_exact":{},"incoming_edges":[{"id":"e-f7365f1dbad216ac","operation":{"operation":"parent_provers","route":"root_domain"},"operation_detail":"root_domain","conclusion_id":"n-fefc87d068bda822","premise_count":0,"theorem":"polytool parent provers, route root_domain: every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign","lean_route":null,"lean_declarations":[],"certificate_hash":"9782df96d451f899548870bcced51364d20a993bc158e2bfad5b8865faad20e1","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[]}],"used_by":[{"id":"e-0bd629cc7b768a78","operation":"basis_map","operation_detail":"A008277","conclusion_id":"n-3933c6a9486f6595","conclusion_name":"basis_map_A008277 of Triangle read by rows, Stirling cycle numbers of order 3, T(n,n) = 1, T(n,k) = 0 if k<0 or k>n, otherwise T(n,k) = T(n-1,k-1)+(n-1)^3*T(n-1,k), for n>=0 and 0<=k<=n."},{"id":"e-0e122e2df114c72b","operation":"basis_map","operation_detail":"A007318","conclusion_id":"n-b096a5459ca672a4","conclusion_name":"basis_map_A007318 of Triangle read by rows, Stirling cycle numbers of order 3, T(n,n) = 1, T(n,k) = 0 if k<0 or k>n, otherwise T(n,k) = T(n-1,k-1)+(n-1)^3*T(n-1,k), for n>=0 and 0<=k<=n."},{"id":"e-8aecb1bbda67a980","operation":"basis_map","operation_detail":"A013611","conclusion_id":"n-2a7a2754f632f0f9","conclusion_name":"basis_map_A013611 of Triangle read by rows, Stirling cycle numbers of order 3, T(n,n) = 1, T(n,k) = 0 if k<0 or k>n, otherwise T(n,k) = T(n-1,k-1)+(n-1)^3*T(n-1,k), for n>=0 and 0<=k<=n."},{"id":"e-e62c7427c0d251dd","operation":"basis_map","operation_detail":"A013609","conclusion_id":"n-7cfac8d939582bff","conclusion_name":"basis_map_A013609 of Triangle read by rows, Stirling cycle numbers of order 3, T(n,n) = 1, T(n,k) = 0 if k<0 or k>n, otherwise T(n,k) = T(n-1,k-1)+(n-1)^3*T(n-1,k), for n>=0 and 0<=k<=n."},{"id":"e-f13e01bf2e1e97ea","operation":"basis_map","operation_detail":"A013610","conclusion_id":"n-c2c5d2e7e07878ae","conclusion_name":"basis_map_A013610 of Triangle read by rows, Stirling cycle numbers of order 3, T(n,n) = 1, T(n,k) = 0 if k<0 or k>n, otherwise T(n,k) = T(n-1,k-1)+(n-1)^3*T(n-1,k), for n>=0 and 0<=k<=n."}],"used_by_total":5,"links":[{"id":"l-21d0622c03598117","kind":"oeis_match","node_a_id":"n-fefc87d068bda822","node_b_id":null,"fingerprint":null,"oeis_id":"A269947","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":9,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":9,"target":"triangle","terms_compared":45},"in_catalog":true,"oeis_id":"A269947","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-2955cb03d68e9421","kind":"oeis_match","node_a_id":"n-fefc87d068bda822","node_b_id":null,"fingerprint":null,"oeis_id":"A249677","in_catalog":0,"target":"reversed_triangle","row_alignment":"tabl","rows_matched":9,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":true,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":9,"target":"reversed_triangle","terms_compared":45},"in_catalog":false,"oeis_id":"A249677","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-a9d307a1bbe9cfd1","kind":"oeis_match","node_a_id":"n-fefc87d068bda822","node_b_id":null,"fingerprint":null,"oeis_id":"A255433","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":14},"in_catalog":false,"oeis_id":"A255433","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A269947","relation":"identification"},{"oeis_id":"A249677","relation":"match_reversed_triangle"},{"oeis_id":"A255433","relation":"match_row_sums"},{"oeis_id":"A269947","relation":"match_triangle"}],"bfiles":[{"node_id":"n-fefc87d068bda822","oeis_id":"A249677","target":"reversed_triangle","oeis_offset":0,"first_row":1,"rows_used":44,"terms_matched":45,"oeis_terms_agreeing":45,"oeis_terms_available":45,"term_count":990,"raw_size":61619},{"node_id":"n-fefc87d068bda822","oeis_id":"A255433","target":"row_sums","oeis_offset":0,"first_row":1,"rows_used":200,"terms_matched":14,"oeis_terms_agreeing":15,"oeis_terms_available":15,"term_count":200,"raw_size":100282},{"node_id":"n-fefc87d068bda822","oeis_id":"A269947","target":"triangle","oeis_offset":0,"first_row":0,"rows_used":44,"terms_matched":45,"oeis_terms_agreeing":45,"oeis_terms_available":45,"term_count":990,"raw_size":57485}],"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: 0,              1;\n  2: 0,              1,              1;\n  3: 0,              8,              9,              1;\n  4: 0,            216,            251,             36,            1;\n  5: 0,          13824,          16280,           2555,          100,           1;\n  6: 0,        1728000,        2048824,         335655,        15055,         225,         1;\n  7: 0,      373248000,      444273984,       74550304,      3587535,       63655,       441,      1;\n  8: 0,   128024064000,   152759224512,    26015028256,   1305074809,    25421200,    214918,    784,    1;\n  9: 0, 65548320768000, 78340747014144, 13472453691584, 694213330464, 14320729209, 135459216, 616326, 1296, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n^3-3*n^2+3*n-1+x)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(n^3 - 3*n^2 + 3*n - 1 + x)*P[n - 1, 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^3 - 3*n^2 + 3*n - 1 + x)*v[i - 1]); 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}},{"oeis_id":"A249677","target":"reversed_triangle","first_row":1,"oeis_offset":0,"reversed":true,"trim_zeros":"low","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0: 1;\n  1: 1,    1;\n  2: 1,    9,       8;\n  3: 1,   36,     251,       216;\n  4: 1,  100,    2555,     16280,        13824;\n  5: 1,  225,   15055,    335655,      2048824,        1728000;\n  6: 1,  441,   63655,   3587535,     74550304,      444273984,       373248000;\n  7: 1,  784,  214918,  25421200,   1305074809,    26015028256,    152759224512,     128024064000;\n  8: 1, 1296,  616326, 135459216,  14320729209,   694213330464,  13472453691584,   78340747014144,    65548320768000;\n  9: 1, 2025, 1561110, 584760870, 113070497673, 11134024923825, 519553971599840, 9899759488178880, 57175952894078976, 47784725839872000;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, down to the lowest nonzero coefficient, where P(n,x) = (n^3+x)*P(n-1,x) for n >= 1, with P(0,x) = x.","mathematica":"P[0, x_] := x;\nP[n_, x_] := P[n, x] = Expand[(n^3 + x)*P[n - 1, x]];\nTable[Reverse[PadRight[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x], n + 1]], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = x; for(n = 1, N, my(i = n + 1); v[i] = (n^3 + x)*v[i - 1]); v};\nrow(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); 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(9)","reading":{"first_row":1,"offset":0,"reversed":true,"trim_zeros":"low","regular_triangle":true}},{"oeis_id":"A269947","target":"triangle","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0: 1;\n  1: 0,              1;\n  2: 0,              1,              1;\n  3: 0,              8,              9,              1;\n  4: 0,            216,            251,             36,            1;\n  5: 0,          13824,          16280,           2555,          100,           1;\n  6: 0,        1728000,        2048824,         335655,        15055,         225,         1;\n  7: 0,      373248000,      444273984,       74550304,      3587535,       63655,       441,      1;\n  8: 0,   128024064000,   152759224512,    26015028256,   1305074809,    25421200,    214918,    784,    1;\n  9: 0, 65548320768000, 78340747014144, 13472453691584, 694213330464, 14320729209, 135459216, 616326, 1296, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n^3-3*n^2+3*n-1+x)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(n^3 - 3*n^2 + 3*n - 1 + x)*P[n - 1, x]];\nTable[PadRight[CoefficientList[P[n, x], x], n + 1], {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^3 - 3*n^2 + 3*n - 1 + x)*v[i - 1]); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); 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":true}}],"derivation":{"root":"n-fefc87d068bda822","steps":[],"edges":[],"text":"A269947"},"headline":"A269947: Triangle read by rows, Stirling cycle numbers of order 3, T(n,n) = 1, T(n,k) = 0 if k<0 or k>n, otherwise T(n,k) = T(n-1,k-1)+(n-1)^3*T(n-1,k), for n>=0 and 0<=k<=n.","chain":null}