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a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"82981d888a9f07705158e1e64401414fb1543ae00a058c62eb34330b0205720f","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-2a67b0774691ba3b","role":"sequence","name":"Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + (j+5)*prime(j)*T(n-2, k-1) with j=6, read by rows.","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:           2;\n  1:          26,           26;\n  2:         338,          676,            8;\n  3:        4394,        13182,         8804,            16;\n  4:       57122,       228488,       238452,         19928,            32;\n  5:      742586,      3712930,      5962904,       3036992,         44496,           64;\n  6:     9653618,     57921708,    118797872,      89046768,      11866960,        98272,         128;\n  7:   125497034,    878479238,   2268032412,    2555369680,    1076770096,     36665472,      215104,       256;\n  8:  1631461442,  13051691536,  39846727484,   56356683400,   34428630480,   5634874592,   101885248,    467328,     512;\n  9: 21208998746, 190880988714, 681370535024, 1206481955840, 1066384479968, 393326420096, 21990604544, 266261504, 1008896, 1024;\n","recurrence_sentence":"The row polynomials satisfy (n-2)*P(n,x) = (143*n-286+(286*n-572)*x)*P(n-2,x) + (2*n-4+(2*n-4)*x)*P(n-1,x) for n >= 3, with P(0,x) = 2, P(1,x) = 26 + 26*x, P(2,x) = 338 + 676*x + 8*x^2.","mathematica":"P[0, x_] := 2;\nP[1, x_] := 26 + 26*x;\nP[2, x_] := 338 + 676*x + 8*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((143*n - 286 + (286*n - 572)*x)*P[n - 2, x] + (2*n - 4 + (2*n - 4)*x)*P[n - 1, x])/(n - 2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 2; v[2] = 26 + 26*x; v[3] = 338 + 676*x + 8*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((143*n - 286 + (286*n - 572)*x)*v[i - 2] + (2*n - 4 + (2*n - 4)*x)*v[i - 1])/(n - 2)); 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-2a67b0774691ba3b","steps":["basis A013609"],"edges":["e-80d83f0e117cb79e"],"text":"A153651 \u2192 basis A013609"},"headline":"A153651 \u2192 basis A013609","chain":null}