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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":"c131f095195cbda94237f8caa39793b931d891d4dee8c8727dcbaeb62e84634f","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-64a8f6054296172d","role":"sequence","name":"basis_map_A008277 of basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A008288) regarded as a triangle. shifted by x -> -x + 0","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,      6,         3;\n  2: 1,     30,        75,          40,            5;\n  3: 1,    126,       903,        1400,          700,           126,              7;\n  4: 1,    510,      9075,       31080,        34755,         15876,           3234,            288,              9;\n  5: 1,   2046,     85503,      583000,      1233650,       1076922,         447909,          95040,          10395,           550,            11;\n  6: 1,   8190,    784875,    10130120,     37542505,      55927872,       40007968,       15196896,        3235518,        393250,         26741,          936,          13;\n  7: 1,  32766,   7125303,   169423800,   1053834600,    2524159638,     2861189331,     1733022720,      604156410,     126626500,      16274258,      1277640,       59150,       1470,        15;\n  8: 1, 131070,  64373475,  2777349160,  28263758255,  105034499388,   179956733502,   163327960224,    85759400727,   27583341500,    5632670758,    744267888,    63832314,    3498600,    117300,    2176,     17;\n  9: 1, 524286, 580344303, 45038667800, 737946423550, 4158489610674, 10450472443873, 13678008027840, 10301531641245, 4772970337850, 1423545395701, 281603415600, 37601709080, 3410085420, 208751670, 8434176, 214149, 3078, 19;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(2*x^2+2*x^3+x^4)*P(n-2,x) - (2*x^2+3*x^3+2*x^4)*d/dx P(n-2,x) - (x^3+x^4)*d^2/dx^2 P(n-2,x) + (1+6*x+2*x^2)*P(n-1,x) + (3*x+2*x^2)*d/dx P(n-1,x) + x^2*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 6*x + 3*x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 6*x + 3*x^2;\nP[n_, x_] := P[n, x] = Expand[-(2*x^2 + 2*x^3 + x^4)*P[n - 2, x] - (2*x^2 + 3*x^3 + 2*x^4)*D[P[n - 2, x], x] - (x^3 + x^4)*D[P[n - 2, x], {x, 2}] + (1 + 6*x + 2*x^2)*P[n - 1, x] + (3*x + 2*x^2)*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 2}]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 6*x + 3*x^2; for(n = 2, N, my(i = n + 1); v[i] = -(2*x^2 + 2*x^3 + x^4)*v[i - 2] - (2*x^2 + 3*x^3 + 2*x^4)*deriv(v[i - 2], x) - (x^3 + x^4)*deriv(deriv(v[i - 2], x), x) + (1 + 6*x + 2*x^2)*v[i - 1] + (3*x + 2*x^2)*deriv(v[i - 1], x) + x^2*deriv(deriv(v[i - 1], x), 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-cdb0db4ad473ee28","steps":["basis A008277","\u03b8+1"],"edges":["e-7ccdf7ec12150e18","e-9e3c5ce457fab2ac"],"text":"basis-map coefficients of A297191, shifted \u2192 basis A008277 \u2192 \u03b8+1"},"headline":"basis-map coefficients of A297191, shifted \u2192 basis A008277 \u2192 \u03b8+1","chain":null}