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binomial(i,j)*12^(i-j)*5^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:             459,             220,              25;\n  1:            9248,            6885,            1650,             125;\n  2:          181781,          184960,           68850,           11000,             625;\n  3:         3507882,         4544525,         2312000,          573750,           68750,           3125;\n  4:        66733279,       105236460,        68167875,        23120000,         4303125,         412500,          15625;\n  5:      1255153588,      2335664765,      1841638050,       795291875,       202300000,       30121875,        2406250,         78125;\n  6:     23389304361,     50206143520,     46713295300,     24555174000,      7952918750,     1618400000,      200812500,      13750000,       390625;\n  7:    432496961342,   1052518696245,   1129638229200,    700699429500,    276245707500,    71576268750,    12138000000,    1290937500,     77343750,     1953125;\n  8:   7945387725299,  21624848067100,  26312967406125,  18827303820000,   8758742868750,  2762457075000,   596468906250,   86700000000,   8068359375,   429687500,    9765625;\n  9: 145151560832328, 436996324891445, 594683321845250, 482404402445625, 258875427525000, 96346171556250, 25322523187500, 4686541406250, 596062500000, 49306640625, 2363281250, 48828125;\n","recurrence_sentence":"The row polynomials satisfy (5*n+22+5*x)*P(n,x) = (85*n+459+(25*n+220)*x+25*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 459 + 220*x + 25*x^2.","mathematica":"P[0, x_] := 459 + 220*x + 25*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((85*n + 459 + (25*n + 220)*x + 25*x^2)*P[n - 1, x])/(5*n + 22 + 5*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 459 + 220*x + 25*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((85*n + 459 + (25*n + 220)*x + 25*x^2)*v[i - 1])/(5*n + 22 + 5*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-a5302200d285579d","steps":["x \u21a6 x+1","D+1"],"edges":["e-bdf366c5a3f3b64d","e-0296474ce0b5c280"],"text":"A038331 \u2192 x \u21a6 x+1 \u2192 D+1"},"headline":"A038331 \u2192 x \u21a6 x+1 \u2192 D+1","chain":null}