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a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"9de2fb87ac8772b0a137df133d04f694f0d3ec6c1eec26c98f7c8070f6a3c327","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-67a75673619cc276","role":"sequence","name":"Square array read by antidiagonals downwards: T(n,k), n>=0, k>=0, in which column 0 is equal to A057427: 0, 1, 1, 1, ..., and for k > 0 column k lists two zeros followed by the partial sums of column 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,   2;\n  1:  1,   4,    4;\n  2:  1,   6,   12,     8;\n  3:  2,  14,   36,    40,    16;\n  4:  3,  26,   88,   144,   112,    32;\n  5:  4,  42,  180,   400,   480,   288,    64;\n  6:  6,  72,  364,  1000,  1600,  1472,   704,   128;\n  7:  9, 122,  712,  2328,  4640,  5728,  4224,  1664,   256;\n  8: 13, 198, 1320,  5040, 12096, 18816, 18816, 11520,  3840,  512;\n  9: 19, 320, 2392, 10424, 29232, 54880, 69440, 57984, 30208, 8704, 1024;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+4*x+4*x^2)*P(n-3,x) + (1+2*x)*P(n-1,x) for n >= 3, with P(0,x) = 1 + 2*x, P(1,x) = 1 + 4*x + 4*x^2, P(2,x) = 1 + 6*x + 12*x^2 + 8*x^3.","mathematica":"P[0, x_] := 1 + 2*x;\nP[1, x_] := 1 + 4*x + 4*x^2;\nP[2, x_] := 1 + 6*x + 12*x^2 + 8*x^3;\nP[n_, x_] := P[n, x] = Expand[(1 + 4*x + 4*x^2)*P[n - 3, x] + (1 + 2*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 + 2*x; v[2] = 1 + 4*x + 4*x^2; v[3] = 1 + 6*x + 12*x^2 + 8*x^3; for(n = 3, N, my(i = n + 1); v[i] = (1 + 4*x + 4*x^2)*v[i - 3] + (1 + 2*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-0d55e0bb3c384646","steps":["reverse","basis A013609"],"edges":["e-0e9e334768ea0a35","e-9d8ee33f10e795ac"],"text":"A102547 \u2192 reverse \u2192 basis A013609"},"headline":"A102547 \u2192 reverse \u2192 basis A013609","chain":null}