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binomial(i,j)*10^(i-j)*3^j.","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:            325,            228,              36;\n  1:           5239,           5850,            2052,             216;\n  2:          81289,         125736,           70200,           16416,            1296;\n  3:        1228123,        2438670,         1886040,          702000,          123120,           7776;\n  4:       18193357,       44212428,        43896060,        22632480,         6318000,         886464,          46656;\n  5:      265474495,      764120994,       928460988,       614544840,       237641040,       53071200,        6205248,        279936;\n  6:     3827659537,    12742775760,     18338903856,     14855375808,      7374538080,     2281353984,      424569600,      42550272,       1679616;\n  7:    54653958307,   206693614998,    344054945520,    330100269408,    200547573408,    79645011264,    20532185856,    3275251200,     287214336,     10077696;\n  8:   774128454229,  3279237498420,   6200808449940,   6881098910400,   4951504041120,  2406570880896,   796450112640,  175990164480,   24564384000,   1914762240,    60466176;\n  9: 10890820856071, 51092477979114, 108214837447860, 136417785898680, 113538132021600, 65359853342784, 26472279689856, 7509386776320, 1451918856960, 180138816000, 12637430784, 362797056;\n","recurrence_sentence":"The row polynomials satisfy (6*n+19+6*x)*P(n,x) = (78*n+325+(36*n+228)*x+36*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 325 + 228*x + 36*x^2.","mathematica":"P[0, x_] := 325 + 228*x + 36*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((78*n + 325 + (36*n + 228)*x + 36*x^2)*P[n - 1, x])/(6*n + 19 + 6*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 325 + 228*x + 36*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((78*n + 325 + (36*n + 228)*x + 36*x^2)*v[i - 1])/(6*n + 19 + 6*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-4397e742863fdbe2","steps":["basis A013609","D+1"],"edges":["e-54bb49968ded72d0","e-f1d31b0874a6263f"],"text":"A038305 \u2192 basis A013609 \u2192 D+1"},"headline":"A038305 \u2192 basis A013609 \u2192 D+1","chain":null}