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Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"certificate_hash":"480281f88269610fb72efff757b93fc6a37b77b7bea06c73201e730dda83048b","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-2b477f2aa77e3efc","role":"sequence","name":"Triangle of coefficients in expansion of (1+x)*(1+3x)*(1+5x)*(1+7x)*...*(1+(2n-1)x).","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,   2;\n  2: 1,   8,    9;\n  3: 1,  18,   69,     60;\n  4: 1,  32,  258,    704,     525;\n  5: 1,  50,  690,   3800,    8445,     5670;\n  6: 1,  72, 1515,  13920,   60695,   117144,     72765;\n  7: 1,  98, 2919,  40180,  286895,  1063986,   1849449,   1081080;\n  8: 1, 128, 5124,  98560, 1040270,  6228096,  20469204,  32785920,  18243225;\n  9: 1, 162, 8388, 214704, 3134670, 27449604, 143993108, 430473312, 645273945, 344594250;\n","recurrence_sentence":"The row polynomials satisfy (2*n-3)*P(n,x) = -(2*n-1+(8*n^2-16*n+6)*x+(8*n^3-28*n^2+30*n-9)*x^2)*P(n-2,x) + (4*n-4+(8*n^2-16*n+6)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(2*n - 1 + (8*n^2 - 16*n + 6)*x + (8*n^3 - 28*n^2 + 30*n - 9)*x^2)*P[n - 2, x] + (4*n - 4 + (8*n^2 - 16*n + 6)*x)*P[n - 1, x])/(2*n - 3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n - 1 + (8*n^2 - 16*n + 6)*x + (8*n^3 - 28*n^2 + 30*n - 9)*x^2)*v[i - 2] + (4*n - 4 + (8*n^2 - 16*n + 6)*x)*v[i - 1])/(2*n - 3)); 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-2b477f2aa77e3efc","steps":["\u03b8+1"],"edges":["e-8fe66dcd63f6963f"],"text":"A109692 \u2192 \u03b8+1"},"headline":"A109692 \u2192 \u03b8+1","chain":null}