{"id":"n-7ca3329c5b61a09d","revision":0,"first_index":0,"fingerprint":"f6-2c26e7d1cc623ede","name":"shift_x_plus_one of basis_map_A013609 of A Riordan array of coordination sequences.","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":false,"rows":[["1"],["5","2"],["23","20","4"],["107","142","60","8"],["497","888","576","160","16"],["2309","5186","4544","1936","400","32"],["10727","29020","32052","18464","5840","960","64"],["49835","157710","210252","152856","65360","16416","2240","128"],["231521","839024","1310624","1151424","621600","210944","43904","5120","256"],["1075589","4391938","7866720","8106016","5291040","2266944","637056","113152","11520","512"],["4996919","22699044","45860676","54233024","41543840","21525888","7635712","1829888","283392","25600","1024"],["23214443","116117006","261224828","348704936","306726560","186574144","80039680","24204544","5054208","693760","56320","2048"],["107848529","588987880","1460113440","2171689824","2157413680","1507415040","759265920","277658624","73135872","13527040","1667072","122880","4096"],["501037445","2966428482","8034338016","13175550640","14590623920","11517663072","6665161088","2861859328","911432448","212569600","35278848","3944448","266240","8192"],["2327695367","14850449516","43628525108","78206174944","95529217648","84088972608","54988598336","27129267200","10144173312","2859996160","598309888","90038272","9211904","573440","16384"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","7","47","317","2137","14407","97127","654797","4414417","29760487","200635007","1352612477","9118849897","61476161767","414451220087"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["1"],["1"]]},"terms":[{"coeff":{"coeffs":[["21","8"],["-13","-4"],["2","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["55","51","10"],["-14","-12","-2"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["21","32","15","2"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":177236,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-74675b5e9adecf87"]},{"id":177237,"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-74675b5e9adecf87"]},{"id":177238,"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-74675b5e9adecf87"]}],"region_exact":{},"incoming_edges":[{"id":"e-74675b5e9adecf87","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-7ca3329c5b61a09d","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":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"e1aba9d9df4cea455f3a55ca8f97e9ce3fc6984072efe024f83d05a43b958034","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}}],"premises":[{"position":0,"node_id":"n-d6be23d68d7706ed","role":"sequence","name":"basis_map_A013609 of A Riordan array of coordination sequences.","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-871d39340033aed7","kind":"candidate_equal","node_a_id":"n-7ca3329c5b61a09d","node_b_id":"n-8b1f92f376305977","fingerprint":null,"oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"reason":"no all-index equality certificate was found","rows_compared":15},"other_id":"n-8b1f92f376305977","other":{"id":"n-8b1f92f376305977","name":"basis_map_A007318 of basis_map_A013609 of A Riordan array of coordination sequences.","first_index":0,"fingerprint":"f6-2c26e7d1cc623ede","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["5","2"],["23","20","4"]],"card_rows":[["1"],["5","2"],["23","20","4"],["107","142","60","8"]],"row_sums_preview":["1","7","47","317","2137","14407","97127","654797"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ec35d1145e9a1211","steps":["basis A013609","basis A007318"],"edges":["e-7383986e8ecff3a4","e-7159b66c7a9f8539"],"text":"A113413 \u2192 basis A013609 \u2192 basis A007318"},"headline":"A113413 \u2192 basis A013609 \u2192 basis A007318","chain":null}}],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:       1;\n  1:       5,       2;\n  2:      23,      20,       4;\n  3:     107,     142,      60,       8;\n  4:     497,     888,     576,     160,      16;\n  5:    2309,    5186,    4544,    1936,     400,      32;\n  6:   10727,   29020,   32052,   18464,    5840,     960,     64;\n  7:   49835,  157710,  210252,  152856,   65360,   16416,   2240,    128;\n  8:  231521,  839024, 1310624, 1151424,  621600,  210944,  43904,   5120,   256;\n  9: 1075589, 4391938, 7866720, 8106016, 5291040, 2266944, 637056, 113152, 11520, 512;\n","recurrence_sentence":"The row polynomials satisfy (n+1)*P(n,x) = (2*n^2-13*n+21-(4*n-8)*x)*P(n-1,x) - (14*n-55+(12*n-51)*x+(2*n-10)*x^2)*d/dx P(n-1,x) + (21+32*x+15*x^2+2*x^3)*d^2/dx^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[((2*n^2 - 13*n + 21 - (4*n - 8)*x)*P[n - 1, x] - (14*n - 55 + (12*n - 51)*x + (2*n - 10)*x^2)*D[P[n - 1, x], x] + (21 + 32*x + 15*x^2 + 2*x^3)*D[P[n - 1, x], {x, 2}])/(n + 1)]];\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] = ((2*n^2 - 13*n + 21 - (4*n - 8)*x)*v[i - 1] - (14*n - 55 + (12*n - 51)*x + (2*n - 10)*x^2)*deriv(v[i - 1], x) + (21 + 32*x + 15*x^2 + 2*x^3)*deriv(deriv(v[i - 1], x), x))/(n + 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-ec35d1145e9a1211","steps":["basis A013609","x \u21a6 x+1"],"edges":["e-7383986e8ecff3a4","e-74675b5e9adecf87"],"text":"A113413 \u2192 basis A013609 \u2192 x \u21a6 x+1"},"headline":"A113413 \u2192 basis A013609 \u2192 x \u21a6 x+1","chain":null}
