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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":"37ca5dc861b72dec683863c8e4f9c9a03aff153aa5cb7cb38118fc84a0ee1745","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-e7434b0161663b3b","role":"sequence","name":"basis_map_A013609 of Triangular array read by rows: row n gives the coefficients of the polynomial p(n,x) defined in Comments.","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:    4,     4;\n  2:    7,     8;\n  3:   19,    36,    12;\n  4:   40,    88,    36;\n  5:   97,   272,   180,   32;\n  6:  217,   696,   552,  128;\n  7:  508,  1900,  1908,  704,   80;\n  8: 1159,  4856,  5652, 2560,  400;\n  9: 2683, 12588, 17076, 9760, 2400, 192;\n","recurrence_sentence":"The row polynomials satisfy (156*n+156+(52*n^2+180*n+128)*x+(8*n^3+32*n^2+40*n+16)*x^2)*P(n,x) = (1092*n+156+(520*n^2+2040*n+128)*x+(56*n^3+592*n^2+1160*n+16)*x^2+(32*n^3+160*n^2+192*n)*x^3)*P(n-1,x) - (1521+(1014*n+4446)*x+(156*n^2+2080*n+4760)*x^2+(200*n^2+1416*n+2224)*x^3+(64*n^2+320*n+384)*x^4)*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[Together[((1092*n + 156 + (520*n^2 + 2040*n + 128)*x + (56*n^3 + 592*n^2 + 1160*n + 16)*x^2 + (32*n^3 + 160*n^2 + 192*n)*x^3)*P[n - 1, x] - (1521 + (1014*n + 4446)*x + (156*n^2 + 2080*n + 4760)*x^2 + (200*n^2 + 1416*n + 2224)*x^3 + (64*n^2 + 320*n + 384)*x^4)*D[P[n - 1, x], x])/(156*n + 156 + (52*n^2 + 180*n + 128)*x + (8*n^3 + 32*n^2 + 40*n + 16)*x^2)]];\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] = ((1092*n + 156 + (520*n^2 + 2040*n + 128)*x + (56*n^3 + 592*n^2 + 1160*n + 16)*x^2 + (32*n^3 + 160*n^2 + 192*n)*x^3)*v[i - 1] - (1521 + (1014*n + 4446)*x + (156*n^2 + 2080*n + 4760)*x^2 + (200*n^2 + 1416*n + 2224)*x^3 + (64*n^2 + 320*n + 384)*x^4)*deriv(v[i - 1], x))/(156*n + 156 + (52*n^2 + 180*n + 128)*x + (8*n^3 + 32*n^2 + 40*n + 16)*x^2)); 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-ac04e52510a48bb5","steps":["basis A013609","\u03b8+1"],"edges":["e-3f48f2d5eb407d05","e-3f7b52207bb619cf"],"text":"A249139 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A249139 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}