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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":"3a0b80ac8d337909c43fcb2302c83d0fc67f46401d648b847f43824b4544c960","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-930be05ae3b321d9","role":"sequence","name":"Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(n) = 4*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,       24;\n  3:    27,      312;\n  4:    81,     2832,       2496;\n  5:   243,    22416,      72768;\n  6:   729,   166632,    1338432,      582144;\n  7:  2187,  1201416,   20054976,    30147072;\n  8:  6561,  8532384,  268661376,   933525504,   241176576;\n  9: 19683, 60146592, 3364829568, 22600046592, 19518763008;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3+(8*n-8)*x)*P(n-1,x) + (4*x-16*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 + (8*n - 8)*x)*P[n - 1, x] + (4*x - 16*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 + (8*n - 8)*x)*v[i - 1] + (4*x - 16*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-930be05ae3b321d9","steps":["\u03b3"],"edges":["e-9e5f717e87031ade"],"text":"A257621 \u2192 \u03b3"},"headline":"A257621 \u2192 \u03b3","chain":null}