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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":"0f0dc47c7fde0dd9bfaa8f35e9032e866b453fc957cb7970f26c1aed4371ecf1","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-05516f61aeaea779","role":"sequence","name":"Triangle read by rows: T(n,k) (1<=k<=n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n-1,k), where m = 5.","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: 1;\n  2: 1,      10;\n  3: 1,      80;\n  4: 1,     510,       800;\n  5: 1,    3100,     19000;\n  6: 1,   18650,    302000,     190000;\n  7: 1,  111960,   4068000,    9080000;\n  8: 1,  671830,  50346000,  267320000,   90800000;\n  9: 1, 4031060, 594115800, 6290960000, 7253200000;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+(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[(1 + (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] = (1 + (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-05516f61aeaea779","steps":["\u03b3"],"edges":["e-a112a311030a6f8e"],"text":"A142460 \u2192 \u03b3"},"headline":"A142460 \u2192 \u03b3","chain":null}