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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"d34f4bbe9593c98b449200a8814d090c64afa383769fed5c1b6e583cf03eccf0","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-a6a633b989da1879","role":"sequence","name":"Triangle whose (i,j)-th entry is binomial(i,j)*7^(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:          1;\n  1:         12,            5;\n  2:        144,          145,           25;\n  3:       1728,         3185,         1275,          125;\n  4:      20736,        62785,        43975,         9750,          625;\n  5:     248832,      1171025,      1281375,       483125,        68750,         3125;\n  6:    2985984,     21151585,     34045375,     19451250,      4615625,       459375,       15625;\n  7:   35831808,    374506865,    854756175,    695410625,    244956250,     40075000,     2953125,      78125;\n  8:  429981696,   6545775745,  20677170175,  23049867750,  11315653125,   2707556250,   324406250,   18437500,    390625;\n  9: 5159780352, 113428096145, 487626622575, 725732280125, 477350238750, 156757846875, 27162843750, 2488593750, 112500000, 1953125;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (12+5*x)*P(n-1,x) + 5*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(12 + 5*x)*P[n - 1, x] + 5*x*D[P[n - 1, x], 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] = (12 + 5*x)*v[i - 1] + 5*x*deriv(v[i - 1], 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-a6a633b989da1879","steps":["basis A008277"],"edges":["e-5070060140385b53"],"text":"A038271 \u2192 basis A008277"},"headline":"A038271 \u2192 basis A008277","chain":null}