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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":"e5d362f7f4d4bc59772d549645a488649bf97de59c08d1e1aae48aa9446392f2","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-6a8ffbe72ba42678","role":"sequence","name":"Triangle T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)*p(n, x)/x, p(n, x) = x*D( p(n-1, x) ), with p(1, x) = x/(1-x)^2, p(2, x) = x*(1+x)/(1-x)^3, and p(3, x) = x*(1+10*x+x^2)/(1-x)^4, read by rows.","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,    8;\n   2: 1,   20;\n   3: 1,   46,     40;\n   4: 1,  100,    304;\n   5: 1,  210,   1512,      608;\n   6: 1,  432,   6216,     8480;\n   7: 1,  878,  22968,    71216,    16960;\n   8: 1, 1772,  79440,   468608,   369664;\n   9: 1, 3562, 263128,  2668832,  4659968,   739328;\n  10: 1, 7144, 846376, 13832864, 44650496, 23075840;\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) for n >= 2, with P(1,x) = 1 + 8*x.","mathematica":"P[1, x_] := 1 + 8*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]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1 + 8*x; for(n = 2, N, my(i = n); v[i] = (1 + 2*n*x)*v[i - 1] + (x - 4*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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-6a8ffbe72ba42678","steps":["\u03b3"],"edges":["e-40e505e18769a2de"],"text":"A166341 \u2192 \u03b3"},"headline":"A166341 \u2192 \u03b3","chain":null}