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(1+3*x+2*x^2)^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:        6,        28,         32;\n  2:       36,       336,       1168,         1792,         1024;\n  3:      216,      3024,      17568,        54208,        93696,         86016,          32768;\n  4:     1296,     24192,     196992,       913920,      2642176,       4874240,        5603328,         3670016,         1048576;\n  5:     7776,    181440,    1900800,     11773440,     47746560,     132471808,      254648320,       334888960,       288358400,       146800640,         33554432;\n  6:    46656,   1306368,   16733952,    129669120,    676961280,    2508484608,     6764990464,     13378584576,     19255787520,     19671285760,      13539213312,       5637144576,       1073741824;\n  7:   279936,   9144576,  138475008,   1288369152,   8227989504,   38155235328,   132490272768,    349962747904,    706614788096,   1085304471552,    1248216481792,    1042401984512,     597537325056,     210453397504,      34359738368;\n  8:  1679616,  62705664, 1095856128,  11900141568,  89873473536,  500542930944,  2126583889920,   7030471655424,  18278334398464,  37495848828928,   60489497313280,   75934216486912,   72715406934016,   51350628990976,   25220047962112,    7696581394432,    1099511627776;\n  9: 10077696, 423263232, 8384643072, 104091402240, 907512201216, 5900519374848, 29650656559104, 117744552640512, 374573897220096, 961743549104128, 1997727451840512, 3349178386219008, 4498114417262592, 4774030095679488, 3916030921408512, 2395560959016960, 1029142883598336, 277076930199552, 35184372088832;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (6+28*x+32*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(6 + 28*x + 32*x^2)*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; for(n = 1, N, my(i = n + 1); v[i] = (6 + 28*x + 32*x^2)*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-eed4f744ce063213","steps":["basis A013611"],"edges":["e-ecef41f482d88baf"],"text":"A200536 \u2192 basis A013611"},"headline":"A200536 \u2192 basis A013611","chain":null}