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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":"a9f6d8390dd5025a03c51e33efec89b42531232da092d80f4d3c64c4406d0227","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-e7968148eadb4da9","role":"sequence","name":"basis_map_A007318 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,    2;\n  2:    7,    4;\n  3:   19,   18,    3;\n  4:   40,   44,    9;\n  5:   97,  136,   45,    4;\n  6:  217,  348,  138,   16;\n  7:  508,  950,  477,   88,   5;\n  8: 1159, 2428, 1413,  320,  25;\n  9: 2683, 6294, 4269, 1220, 150, 6;\n","recurrence_sentence":"The row polynomials satisfy (78*n+78+(13*n^2+45*n+32)*x+(n^3+4*n^2+5*n+2)*x^2)*P(n,x) = (546*n+78+(130*n^2+510*n+32)*x+(7*n^3+74*n^2+145*n+2)*x^2+(2*n^3+10*n^2+12*n)*x^3)*P(n-1,x) - (1521+(507*n+2223)*x+(39*n^2+520*n+1190)*x^2+(25*n^2+177*n+278)*x^3+(4*n^2+20*n+24)*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[((546*n + 78 + (130*n^2 + 510*n + 32)*x + (7*n^3 + 74*n^2 + 145*n + 2)*x^2 + (2*n^3 + 10*n^2 + 12*n)*x^3)*P[n - 1, x] - (1521 + (507*n + 2223)*x + (39*n^2 + 520*n + 1190)*x^2 + (25*n^2 + 177*n + 278)*x^3 + (4*n^2 + 20*n + 24)*x^4)*D[P[n - 1, x], x])/(78*n + 78 + (13*n^2 + 45*n + 32)*x + (n^3 + 4*n^2 + 5*n + 2)*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] = ((546*n + 78 + (130*n^2 + 510*n + 32)*x + (7*n^3 + 74*n^2 + 145*n + 2)*x^2 + (2*n^3 + 10*n^2 + 12*n)*x^3)*v[i - 1] - (1521 + (507*n + 2223)*x + (39*n^2 + 520*n + 1190)*x^2 + (25*n^2 + 177*n + 278)*x^3 + (4*n^2 + 20*n + 24)*x^4)*deriv(v[i - 1], x))/(78*n + 78 + (13*n^2 + 45*n + 32)*x + (n^3 + 4*n^2 + 5*n + 2)*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 A007318","\u03b8+1"],"edges":["e-16890f4567488209","e-c5cf17a3b1c33940"],"text":"A249139 \u2192 basis A007318 \u2192 \u03b8+1"},"headline":"A249139 \u2192 basis A007318 \u2192 \u03b8+1","chain":null}