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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":"3a7787e2d5dc12934b8d70718a1873410aff2110c877307f0eda70463b986fcd","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-09153823987c7e65","role":"sequence","name":"Triangle read by rows, related to number of permutations of [n] with 0 successions and k rises.","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: 1,    5;\n   2: 1,   18;\n   3: 1,   47,      61;\n   4: 1,  108,     479;\n   5: 1,  233,    2414,     1385;\n   6: 1,  486,    9970,    19028;\n   7: 1,  995,   36867,   157658,     50521;\n   8: 1, 2016,  127401,  1025268,   1073517;\n   9: 1, 4061,  421372,  5780619,  13253431,  2702765;\n  10: 1, 8154, 1353284, 29707752, 124815323, 82112518;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+2*n*x)*P(n-1,x) + (x-4*x^2)*d/dx P(n-1,x) + (n+2)*x*P(n-2,x) for n >= 3, with P(1,x) = 1 + 5*x, P(2,x) = 1 + 18*x.","mathematica":"P[1, x_] := 1 + 5*x;\nP[2, x_] := 1 + 18*x;\nP[n_, x_] := P[n, x] = Expand[(1 + 2*n*x)*P[n - 1, x] + (x - 4*x^2)*D[P[n - 1, x], x] + (n + 2)*x*P[n - 2, x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); 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