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Gamma(c z) with c != 0 has the roots of Gamma divided by c","lean_route":null,"lean_declarations":[],"certificate_hash":"c8d0b5b0e721b42196cebb53324838fe7f86827e537d1bd361ab8badfb0a5f50","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-b58ba5723c3a3a3f","role":"sequence","name":"Triangle, read by rows, T(n,k) = t(n-k, k) where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(x) = x + 3.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:      9,        6;\n   2:     27,       60;\n   3:     81,      402,       120;\n   4:    243,     2256,      2208;\n   5:    729,    11454,     24576,       4416;\n   6:   2187,    54564,    214512,     124800;\n   7:   6561,   248874,   1618200,    2035872,     249600;\n   8:  19683,  1100472,  11077488,   25160832,    9890688;\n   9:  59049,  4756182,  70794048,  261739872,  220199808,   19781376;\n  10: 177147, 20205708, 430069152, 2419967808, 3635317632, 1039050240;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3+2*n*x)*P(n-1,x) + (x-4*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 9 + 6*x.","mathematica":"P[1, x_] := 9 + 6*x;\nP[n_, x_] := P[n, x] = Expand[(3 + 2*n*x)*P[n - 1, x] + (x - 4*x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); 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