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(1+11x)^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:            672,            1496,              484;\n  1:          11232,           44352,            49368,             10648;\n  2:         172800,          988416,          1951488,           1448128,            234256;\n  3:        2529792,        19008000,         54362880,          71554560,          39823520,           5153632;\n  4:       35831808,       333932544,       1254528000,        2391966720,        2361300480,        1051340928,          113379904;\n  5:      495673344,      5518098432,      25712805888,       64399104000,       92090718720,       72728054784,        26984417152,         2494357888;\n  6:     6736379904,     87238508544,     485592662016,     1508484612096,     2833560576000,     3241593298944,      2133356273664,       678465345536,        54875873536;\n  7:    90296156160,   1333803220992,    8636612345856,    32049115693056,    74669988298752,   112208998809600,    106972578865152,     60343506026496,     16792017302016,     1207269217792;\n  8:  1197069041664,  19865154355200,  146718354309120,   633351572029440,  1762701363118080,  3285479485145088,   4114329956352000,   3361995335761920,   1659446415728640,   410471534049280,   26559922791424;\n  9: 15727010512896, 289690708082688, 2403683676979200, 11835280580935680, 38317770107781120, 85314745974915072, 132514339234185216, 142238264205312000, 101700358906798080, 44620670289592320, 9933411123992576, 584318301411328;\n","recurrence_sentence":"The row polynomials satisfy (11*n+17+11*x)*P(n,x) = (132*n+336+(242*n+748)*x+242*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 672 + 1496*x + 484*x^2.","mathematica":"P[0, x_] := 672 + 1496*x + 484*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((132*n + 336 + (242*n + 748)*x + 242*x^2)*P[n - 1, x])/(11*n + 17 + 11*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 672 + 1496*x + 484*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((132*n + 336 + (242*n + 748)*x + 242*x^2)*v[i - 1])/(11*n + 17 + 11*x)); 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-ec89fb3fc20684fc","steps":["x \u21a6 x/(x+1)","D+1"],"edges":["e-492aa71fabef2dc6","e-b5a389570b880716"],"text":"A038336 \u2192 x \u21a6 x/(x+1) \u2192 D+1"},"headline":"A038336 \u2192 x \u21a6 x/(x+1) \u2192 D+1","chain":null}