{"id":"n-61bf3ec06b6c88be","revision":1,"first_index":0,"fingerprint":"f6-666ae795e3b08a3e","name":"basis_map_A007318 of Triangle read by rows, s_4(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.","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","1"],["32","12","1"],["384","176","24","1"],["6144","3200","560","40","1"],["122880","70144","14400","1360","60","1"],["2949120","1806336","415744","47040","2800","84","1"],["82575360","53526528","13447168","1732864","125440","5152","112","1"],["2642411520","1795424256","483835904","68898816","5746944","290304","8736","144","1"],["95126814720","67277684736","19213516800","2964193280","275788800","16197888","604800","13920","180","1"],["3805072588800","2786234204160","835818356736","137781248000","13995745280","923704320","40389888","1161600","21120","220","1"],["167423193907200","126399377571840","39562241900544","6898193268736","753594040320","54638735360","2700859392","91500288","2090880","30800","264","1"],["8036313307545600","6234593317355520","2025386988797952","370675518799872","43070707204096","3376253337600","184279986176","7092873216","191862528","3569280","43472","312","1"],["417888291992371200","332235165810032640","111554716734849024","21300513966391296","2610352293412864","218635880759296","12958812618752","553109393408","17069724672","377465088","5829824","59696","364","1"],["23401744351572787200","19023057577354199040","6579299302961577984","1304383498852761600","167480242397511680","14853961615933440","944329387409408","43932938649600","1509013975040","38207769600","703935232","9172800","80080","420","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","45","585","9945","208845","5221125","151412625","4996616625","184874815125","7579867420125","341094033905625","16713607661375625","885821206052908125","50491808745015763125"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A007318","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","1"],["4","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":161990,"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-2dc9098c1ed2fab8"]},{"id":161991,"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-2dc9098c1ed2fab8"]},{"id":161992,"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-2dc9098c1ed2fab8"]}],"region_exact":{},"incoming_edges":[{"id":"e-2dc9098c1ed2fab8","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-61bf3ec06b6c88be","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"0006d09d5d8911f755edf64d9ecec99e79c4f068292d668ca8e861afad652af8","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-9c5c208df8605708","role":"sequence","name":"Triangle read by rows, s_4(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.","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-227139d97b981bce","kind":"oeis_match","node_a_id":"n-61bf3ec06b6c88be","node_b_id":null,"fingerprint":null,"oeis_id":"A007696","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":"A007696","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A007696","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,           1;\n  2:          32,          12,           1;\n  3:         384,         176,          24,          1;\n  4:        6144,        3200,         560,         40,         1;\n  5:      122880,       70144,       14400,       1360,        60,        1;\n  6:     2949120,     1806336,      415744,      47040,      2800,       84,      1;\n  7:    82575360,    53526528,    13447168,    1732864,    125440,     5152,    112,     1;\n  8:  2642411520,  1795424256,   483835904,   68898816,   5746944,   290304,   8736,   144,   1;\n  9: 95126814720, 67277684736, 19213516800, 2964193280, 275788800, 16197888, 604800, 13920, 180, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (4*n+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[(4*n + 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] = (4*n + 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}}],"derivation":{"root":"n-9c5c208df8605708","steps":["basis A007318"],"edges":["e-2dc9098c1ed2fab8"],"text":"A225471 \u2192 basis A007318"},"headline":"A225471 \u2192 basis A007318","chain":null}
