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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":"93616604c3a9be7a38d378052cf71346f68550c6e3ef556348668815fc36aca8","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-da6c9018f05787b2","role":"sequence","name":"basis_map_A013609 of Riordan array (1/(1-2*x), x*(1+x)/(1-2*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,      4;\n  2:    10,     28,      12;\n  3:    33,    140,     132,      32;\n  4:   109,    620,     912,     480,      80;\n  5:   360,   2568,    5148,    4256,    1520,     192;\n  6:  1189,  10200,   25920,   29408,   16480,    4416,     448;\n  7:  3927,  39364,  121212,  175328,  135120,   56640,   12096,   1024;\n  8: 12970, 148756,  538248,  947744,  933680,  538176,  179200,  31744,  2304;\n  9: 42837, 553220, 2300316, 4777120, 5743840, 4236288, 1937600, 533504, 80640, 5120;\n","recurrence_sentence":"The row polynomials satisfy -(52*n+(52*n^2-84*n)*x+(24*n^3-136*n^2-288*n)*x^2-(48*n^3+128*n^2+80*n)*x^3)*P(n,x) = -(416*n-260+(572*n^2+836*n-1296)*x+(192*n^3+976*n^2-352*n-2144)*x^2+(192*n^3+80*n^2-1296*n-1344)*x^3-(192*n^2+512*n+320)*x^4)*P(n-1,x) + (169+(338*n+1066)*x+(156*n^2+1612*n+2488)*x^2+(552*n^2+2616*n+2608)*x^3+(528*n^2+1616*n+1168)*x^4+(96*n^2+256*n+160)*x^5)*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[(-(416*n - 260 + (572*n^2 + 836*n - 1296)*x + (192*n^3 + 976*n^2 - 352*n - 2144)*x^2 + (192*n^3 + 80*n^2 - 1296*n - 1344)*x^3 - (192*n^2 + 512*n + 320)*x^4)*P[n - 1, x] + (169 + (338*n + 1066)*x + (156*n^2 + 1612*n + 2488)*x^2 + (552*n^2 + 2616*n + 2608)*x^3 + (528*n^2 + 1616*n + 1168)*x^4 + (96*n^2 + 256*n + 160)*x^5)*D[P[n - 1, x], x])/(-52*n - (52*n^2 - 84*n)*x - (24*n^3 - 136*n^2 - 288*n)*x^2 + (48*n^3 + 128*n^2 + 80*n)*x^3)]];\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] = (-(416*n - 260 + (572*n^2 + 836*n - 1296)*x + (192*n^3 + 976*n^2 - 352*n - 2144)*x^2 + (192*n^3 + 80*n^2 - 1296*n - 1344)*x^3 - (192*n^2 + 512*n + 320)*x^4)*v[i - 1] + (169 + (338*n + 1066)*x + (156*n^2 + 1612*n + 2488)*x^2 + (552*n^2 + 2616*n + 2608)*x^3 + (528*n^2 + 1616*n + 1168)*x^4 + (96*n^2 + 256*n + 160)*x^5)*deriv(v[i - 1], x))/(-52*n - (52*n^2 - 84*n)*x - (24*n^3 - 136*n^2 - 288*n)*x^2 + (48*n^3 + 128*n^2 + 80*n)*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-d5c14b9fec875ecc","steps":["basis A013609","\u03b8+1"],"edges":["e-eac85dfbd4ac8581","e-6c1397d4d10379a0"],"text":"A121574 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A121574 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}