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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":"ac9b96f13307baca4661997a5fd91150980225ceff7889c45efd1adbe4a3810d","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-c47b645304a2946e","role":"sequence","name":"Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = x + 4.","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:      4;\n  2:     16,       8;\n  3:     64,     104;\n  4:    256,     904,      208;\n  5:   1024,    6568,     4864;\n  6:   4096,   43080,    68592,      9728;\n  7:  16384,  264552,   756192,    342464;\n  8:  65536, 1552136,  7182672,   6934400,   684928;\n  9: 262144, 8809256, 61721664, 106002176, 33217024;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (4+(2*n-2)*x)*P(n-1,x) + (x-4*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[(4 + (2*n - 2)*x)*P[n - 1, x] + (x - 4*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)); 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