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a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"1e1e8de4d6f4be5e47db559fadac1eaa05f3313fcdf6ef309df51c835a40ec3a","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-a3274460eb3b9452","role":"sequence","name":"Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + ((n+1)*(n+2)*(2*n+3)/6)*T(n-2, k-1), 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:          1;\n  1:          2,           4;\n  2:         18,          72,           16;\n  3:         96,         576,          800,           64;\n  4:       1182,        9456,        20624,         6848,          256;\n  5:      11100,      111000,       361536,       393216,        51200,         1024;\n  6:     187680,     2252160,      9349792,     14740736,      5545984,       350208,         4096;\n  7:    2639760,    36956640,    192037568,    442110080,    401363968,     64872448,      2244608,      16384;\n  8:   58768320,   940293120,   5764054816,  16512243072,  20956212736,   8157431808,    664383488,   13680640,    65536;\n  9: 1133844240, 20409196320, 146136971392, 522030940928, 943336048640, 743216117760, 134726238208, 6147899392, 80084992, 262144;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = ((1/3)*n^3+(3/2)*n^2+(13/6)*n+1+((4/3)*n^3+6*n^2+(26/3)*n+4)*x)*P(n-2,x) + (2+4*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 4*x;\nP[n_, x_] := P[n, x] = Expand[((1/3)*n^3 + (3/2)*n^2 + (13/6)*n + 1 + ((4/3)*n^3 + 6*n^2 + (26/3)*n + 4)*x)*P[n - 2, x] + (2 + 4*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] = 2 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((1/3)*n^3 + (3/2)*n^2 + (13/6)*n + 1 + ((4/3)*n^3 + 6*n^2 + (26/3)*n + 4)*x)*v[i - 2] + (2 + 4*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-a3274460eb3b9452","steps":["basis A013611"],"edges":["e-1820edefae4adbcf"],"text":"A154228 \u2192 basis A013611"},"headline":"A154228 \u2192 basis A013611","chain":null}