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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":"d70ea6af8ba00a00d27283c08181f4e8da2767466515a0472af8a7e648ca6621","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-0c5ba5f813be2bea","role":"sequence","name":"shift_x_plus_one of Triangle of B-analogs of Stirling numbers of 2nd kind.","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:       2,        2;\n  2:       6,       12,        3;\n  3:      24,       76,       48,         4;\n  4:     116,      512,      552,       176,         5;\n  5:     648,     3688,     5688,      3296,       630,        6;\n  6:    4088,    28592,    57708,     49056,     18030,     2220,        7;\n  7:   28640,   238320,   597264,    675216,    374680,    94788,     7700,       8;\n  8:  219920,  2126336,  6386880,   9062784,   6806280,  2688960,   491232,   26304,     9;\n  9: 1832224, 20214304, 70928832, 121522432, 115817840, 63077232, 18690224, 2543872, 88650, 10;\n","recurrence_sentence":"The row polynomials satisfy -(2*x-(2*n-1)*x^3)*P(n,x) = (16*n-16+(8*n^2+24*n-44)*x-16*x^2-(8*n^3+20*n^2-16*n-28)*x^3-(16*n^3-28*n-12)*x^4-(8*n^3-12*n^2-2*n+3)*x^5)*P(n-2,x) - (8+(8*n+24)*x+(16*n+16)*x^2-(8*n^2+4*n+12)*x^3-(24*n^2+24*n+14)*x^4-(24*n^2+12*n)*x^5-(8*n^2-2)*x^6)*d/dx P(n-2,x) - (8+(8*n+4)*x-8*x^2-(8*n^2+4*n+2)*x^3-(8*n^2-8*n+2)*x^4)*P(n-1,x) + (4*x+8*x^2-(4*n-6)*x^3-(8*n-4)*x^4-(4*n-2)*x^5)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[((16*n - 16 + (8*n^2 + 24*n - 44)*x - 16*x^2 - (8*n^3 + 20*n^2 - 16*n - 28)*x^3 - (16*n^3 - 28*n - 12)*x^4 - (8*n^3 - 12*n^2 - 2*n + 3)*x^5)*P[n - 2, x] - (8 + (8*n + 24)*x + (16*n + 16)*x^2 - (8*n^2 + 4*n + 12)*x^3 - (24*n^2 + 24*n + 14)*x^4 - (24*n^2 + 12*n)*x^5 - (8*n^2 - 2)*x^6)*D[P[n - 2, x], x] - (8 + (8*n + 4)*x - 8*x^2 - (8*n^2 + 4*n + 2)*x^3 - (8*n^2 - 8*n + 2)*x^4)*P[n - 1, x] + (4*x + 8*x^2 - (4*n - 6)*x^3 - (8*n - 4)*x^4 - (4*n - 2)*x^5)*D[P[n - 1, x], x])/(-2*x + (2*n - 1)*x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((16*n - 16 + (8*n^2 + 24*n - 44)*x - 16*x^2 - (8*n^3 + 20*n^2 - 16*n - 28)*x^3 - (16*n^3 - 28*n - 12)*x^4 - (8*n^3 - 12*n^2 - 2*n + 3)*x^5)*v[i - 2] - (8 + (8*n + 24)*x + (16*n + 16)*x^2 - (8*n^2 + 4*n + 12)*x^3 - (24*n^2 + 24*n + 14)*x^4 - (24*n^2 + 12*n)*x^5 - (8*n^2 - 2)*x^6)*deriv(v[i - 2], x) - (8 + (8*n + 4)*x - 8*x^2 - (8*n^2 + 4*n + 2)*x^3 - (8*n^2 - 8*n + 2)*x^4)*v[i - 1] + (4*x + 8*x^2 - (4*n - 6)*x^3 - (8*n - 4)*x^4 - (4*n - 2)*x^5)*deriv(v[i - 1], x))/(-2*x + (2*n - 1)*x^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-28595a60eb57b37f","steps":["x \u21a6 x+1","\u03b8+1"],"edges":["e-4c7248607accd612","e-25cd3f49384cc5a7"],"text":"A039756 \u2192 x \u21a6 x+1 \u2192 \u03b8+1"},"headline":"A039756 \u2192 x \u21a6 x+1 \u2192 \u03b8+1","chain":null}