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a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"546c9c294cef948e963165cd129fd0fcb82d9581fd5fdd148cc4bbaa4057c09d","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-71e25d00553f2c01","role":"sequence","name":"Triangle T(n, k, j) =  T(n-1, k, j) + T(n-1, k-1, j) + (2*j + 1)*prime(j)*T(n-2, k-1, j) with T(2, k, j) = prime(j) and j = 8, 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:           38,            19;\n  2:          722,           722,             2;\n  3:        13718,         20577,          6863,             2;\n  4:       260642,        521284,        268155,          7513,             2;\n  5:      4952198,      12380495,       9920714,       2500576,          8163,            2;\n  6:     94091762,     282275286,     287210720,     103962630,       4944247,         8813,          2;\n  7:   1787743478,    6257102173,    8059987233,    4507212650,     924173821,      7599168,       9463,        2;\n  8:  33967126082,  135868504328,  206321056577,  143423404583,   41532481563,   2539210537,   10465339,    10113,     2;\n  9: 645375395558, 2904189280011, 5172930495620, 4552373427952, 1980826197842, 347573578084, 5017729028, 13542760, 10763, 2;\n","recurrence_sentence":"The row polynomials satisfy (n-2)*P(n,x) = (323*n-646+(323*n-646)*x)*P(n-2,x) + (2*n-4+(n-2)*x)*P(n-1,x) for n >= 3, with P(0,x) = 2, P(1,x) = 38 + 19*x, P(2,x) = 722 + 722*x + 2*x^2.","mathematica":"P[0, x_] := 2;\nP[1, x_] := 38 + 19*x;\nP[2, x_] := 722 + 722*x + 2*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((323*n - 646 + (323*n - 646)*x)*P[n - 2, x] + (2*n - 4 + (n - 2)*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] = 38 + 19*x; v[3] = 722 + 722*x + 2*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((323*n - 646 + (323*n - 646)*x)*v[i - 2] + (2*n - 4 + (n - 2)*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-71e25d00553f2c01","steps":["basis A007318"],"edges":["e-50634343b1f85808"],"text":"A153653 \u2192 basis A007318"},"headline":"A153653 \u2192 basis A007318","chain":null}