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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":"240faf5833439ee3f50fb8a7b663b2333bff1d3ce71002af9adea52d4cd04b44","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-fb2cd7d8c683bee9","role":"sequence","name":"shift_x_plus_one of Riordan array (1+2x,x(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:   3,    2;\n  2:   4,   10,    3;\n  3:   7,   26,   21,     4;\n  4:  11,   58,   78,    36,     5;\n  5:  18,  120,  225,   172,    55,     6;\n  6:  29,  236,  570,   612,   320,    78,    7;\n  7:  47,  450, 1329,  1844,  1355,   534,  105,    8;\n  8:  76,  838, 2928,  4988,  4745,  2622,  826,  136,   9;\n  9: 123, 1534, 6189, 12508, 14640, 10476, 4613, 1208, 171, 10;\n","recurrence_sentence":"The row polynomials satisfy (10*n+(5*n^2+9*n)*x+(n^3+2*n^2+n)*x^2)*P(n,x) = -(20*n-50+(15*n^2-27*n-50)*x+(2*n^3-n^2-23*n-18)*x^2-(2*n^2+4*n+2)*x^3)*P(n-1,x) + (25+(25*n+50)*x+(5*n^2+40*n+34)*x^2+(6*n^2+17*n+10)*x^3+(n^2+2*n+1)*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[(-(20*n - 50 + (15*n^2 - 27*n - 50)*x + (2*n^3 - n^2 - 23*n - 18)*x^2 - (2*n^2 + 4*n + 2)*x^3)*P[n - 1, x] + (25 + (25*n + 50)*x + (5*n^2 + 40*n + 34)*x^2 + (6*n^2 + 17*n + 10)*x^3 + (n^2 + 2*n + 1)*x^4)*D[P[n - 1, x], x])/(10*n + (5*n^2 + 9*n)*x + (n^3 + 2*n^2 + n)*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] = (-(20*n - 50 + (15*n^2 - 27*n - 50)*x + (2*n^3 - n^2 - 23*n - 18)*x^2 - (2*n^2 + 4*n + 2)*x^3)*v[i - 1] + (25 + (25*n + 50)*x + (5*n^2 + 40*n + 34)*x^2 + (6*n^2 + 17*n + 10)*x^3 + (n^2 + 2*n + 1)*x^4)*deriv(v[i - 1], x))/(10*n + (5*n^2 + 9*n)*x + (n^3 + 2*n^2 + n)*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-c27b11147fce8bfa","steps":["basis A007318","\u03b8+1"],"edges":["e-3cf676fe863061a7","e-a05676ae98c120a5"],"text":"A113214 \u2192 basis A007318 \u2192 \u03b8+1"},"headline":"A113214 \u2192 basis A007318 \u2192 \u03b8+1","chain":null}