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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":"9f20d190ebc4fd5192616a334f278d95251856e2e63fe2dd8e3e93e9a5dce8b0","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-4155b181fd90d601","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(n) = 5*n + 3.","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:     3;\n  2:     9,        30;\n  3:    27,       420;\n  4:    81,      4170,        4200;\n  5:   243,     36600,      138000;\n  6:   729,    304950,     2892000,     1380000;\n  7:  2187,   2483340,    49794000,    82680000;\n  8:  6561,  20019810,   771489000,  2982060000,   826800000;\n  9: 19683, 160683360, 11230545600, 84536640000, 78657600000;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3+(10*n-10)*x)*P(n-1,x) + (5*x-20*x^2)*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[(3 + (10*n - 10)*x)*P[n - 1, x] + (5*x - 20*x^2)*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] = (3 + (10*n - 10)*x)*v[i - 1] + (5*x - 20*x^2)*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-4155b181fd90d601","steps":["\u03b3"],"edges":["e-b6ca1e00a1efa813"],"text":"A257623 \u2192 \u03b3"},"headline":"A257623 \u2192 \u03b3","chain":null}