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a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"65bc44a833a154f0ab6ca63478ae18a33bc5b5361e97ffbace97299980cbe3b6","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-787407680a56b31c","role":"sequence","name":"Triangle read by rows: T(n,k) = k*(k+1)*binomial(n,k)/2 (1 <= k <= n).","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:     5,     12;\n  2:    18,     84,      96;\n  3:    56,    384,     864,      640;\n  4:   160,   1440,    4800,     7040,     3840;\n  5:   432,   4800,   21120,    46080,    49920,    21504;\n  6:  1120,  14784,   80640,   232960,   376320,   322560,    114688;\n  7:  2816,  43008,  279552,  1003520,  2150400,  2752512,   1949696,    589824;\n  8:  6912, 119808,  903168,  3870720, 10321920, 17547264,  18579456,  11206656,  2949120;\n  9: 16640, 322560, 2764800, 13762560, 43868160, 92897280, 130744320, 117964800, 61931520, 14417920;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (8+48*x+96*x^2+64*x^3)*P(n-3,x) - (12+48*x+48*x^2)*P(n-2,x) + (6+12*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 5 + 12*x, P(2,x) = 18 + 84*x + 96*x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 5 + 12*x;\nP[2, x_] := 18 + 84*x + 96*x^2;\nP[n_, x_] := P[n, x] = Expand[(8 + 48*x + 96*x^2 + 64*x^3)*P[n - 3, x] - (12 + 48*x + 48*x^2)*P[n - 2, x] + (6 + 12*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; v[2] = 5 + 12*x; v[3] = 18 + 84*x + 96*x^2; for(n = 3, N, my(i = n + 1); v[i] = (8 + 48*x + 96*x^2 + 64*x^3)*v[i - 3] - (12 + 48*x + 48*x^2)*v[i - 2] + (6 + 12*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-787407680a56b31c","steps":["basis A013611"],"edges":["e-7913bd6b746d709c"],"text":"A124932 \u2192 basis A013611"},"headline":"A124932 \u2192 basis A013611","chain":null}