{"id":"n-60109662d824f4a5","revision":0,"first_index":0,"fingerprint":"f6-f3a404a357e5fc2c","name":"basis_map_A007318 of Triangle read by rows: T(n, k) = binomial(n, k) if n < 3, otherwise T(n, k) = binomial(n, k) + 2*binomial(n-2, k-1).","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"],["2","1"],["4","4","1"],["12","18","8","1"],["24","48","34","10","1"],["48","120","116","54","12","1"],["96","288","352","224","78","14","1"],["192","672","992","800","380","106","16","1"],["384","1536","2656","2592","1560","592","138","18","1"],["768","3456","6848","7840","5712","2744","868","174","20","1"],["1536","7680","17152","22528","19264","11200","4480","1216","214","22","1"],["3072","16896","41984","62208","61056","41664","20160","6912","1644","258","24","1"],["6144","36864","100864","166400","184320","144384","81984","33984","10200","2160","306","26","1"],["12288","79872","238592","433664","535040","473088","308352","149952","54384","14520","2772","358","28","1"],["24576","172032","557056","1105920","1503744","1481216","1089792","608256","258720","83424","20064","3488","414","30","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","9","39","117","351","1053","3159","9477","28431","85293","255879","767637","2302911","6908733"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A007318","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2","1"],["4","4","1"],["12","18","8","1"]],"recurrence":{"denominator":{"coeffs":[["-3"],["1"]]},"terms":[{"coeff":{"coeffs":[["-6","-3"],["2","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":195216,"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-72dddb5f81e709cd"]},{"id":195217,"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-72dddb5f81e709cd"]},{"id":195218,"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-72dddb5f81e709cd"]},{"id":195219,"position":3,"claim":"interlacing","from_index":4,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":4,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 4","region_exact":null,"edges":["e-72dddb5f81e709cd"]}],"region_exact":{},"incoming_edges":[{"id":"e-72dddb5f81e709cd","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-60109662d824f4a5","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":"603614fbaed526022f5630dc6a8f5328faa2f961e34278849f4049cca6027c8f","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":4,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-fc7cbd97dfcf0d00","role":"sequence","name":"Triangle read by rows: T(n, k) = binomial(n, k) if n < 3, otherwise T(n, k) = binomial(n, k) + 2*binomial(n-2, k-1).","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":[],"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:   2,    1;\n  2:   4,    4,    1;\n  3:  12,   18,    8,    1;\n  4:  24,   48,   34,   10,    1;\n  5:  48,  120,  116,   54,   12,    1;\n  6:  96,  288,  352,  224,   78,   14,   1;\n  7: 192,  672,  992,  800,  380,  106,  16,   1;\n  8: 384, 1536, 2656, 2592, 1560,  592, 138,  18,  1;\n  9: 768, 3456, 6848, 7840, 5712, 2744, 868, 174, 20, 1;\n","recurrence_sentence":"The row polynomials satisfy (n-3)*P(n,x) = (2*n-6+(n-3)*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2 + x, P(2,x) = 4 + 4*x + x^2, P(3,x) = 12 + 18*x + 8*x^2 + x^3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + x;\nP[2, x_] := 4 + 4*x + x^2;\nP[3, x_] := 12 + 18*x + 8*x^2 + x^3;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n - 6 + (n - 3)*x)*P[n - 1, x])/(n - 3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + x; v[3] = 4 + 4*x + x^2; v[4] = 12 + 18*x + 8*x^2 + x^3; for(n = 4, N, my(i = n + 1); v[i] = ((2*n - 6 + (n - 3)*x)*v[i - 1])/(n - 3)); 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-fc7cbd97dfcf0d00","steps":["basis A007318"],"edges":["e-72dddb5f81e709cd"],"text":"A147644 \u2192 basis A007318"},"headline":"A147644 \u2192 basis A007318","chain":null}
