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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":"0d4f7ed3b5a53d35dc14a422fa2ed7e6fee4baa6d30b544183e0c42dabdce065","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-a9307f854ca152e2","role":"sequence","name":"basis_map_A007318 of Riordan array (1, 3+x).","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:     1;\n   2:     3,      6;\n   3:    10,     38,      27;\n   4:    33,    186,     261,      108;\n   5:   109,    814,    1704,     1404,      405;\n   6:   360,   3348,    9315,    11484,     6615,     1458;\n   7:  1189,  13236,   45936,    75388,    65070,    28674,     5103;\n   8:  3927,  50910,  211707,   433812,   498885,   329670,   117369,   17496;\n   9: 12970, 191906,  930006,  2284900,  3282755,  2891754,  1544508,  460728,   59049;\n  10: 42837, 712302, 3941667, 11290812, 19457790, 21421512, 15256647, 6829272, 1751787, 196830;\n","recurrence_sentence":"The row polynomials satisfy (78*n-78+(39*n^2+33*n-72)*x+(3*n^3+27*n^2-30*n)*x^2)*P(n,x) = -(52*n-338+(39*n^2-277*n-520)*x+(2*n^3+7*n^2-481*n-198)*x^2-(18*n^2+180*n)*x^3)*P(n-1,x) + (169+(169*n+429)*x+(13*n^2+416*n+359)*x^2+(22*n^2+337*n+99)*x^3+(9*n^2+90*n)*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(52*n - 338 + (39*n^2 - 277*n - 520)*x + (2*n^3 + 7*n^2 - 481*n - 198)*x^2 - (18*n^2 + 180*n)*x^3)*P[n - 1, x] + (169 + (169*n + 429)*x + (13*n^2 + 416*n + 359)*x^2 + (22*n^2 + 337*n + 99)*x^3 + (9*n^2 + 90*n)*x^4)*D[P[n - 1, x], x])/(78*n - 78 + (39*n^2 + 33*n - 72)*x + (3*n^3 + 27*n^2 - 30*n)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(52*n - 338 + (39*n^2 - 277*n - 520)*x + (2*n^3 + 7*n^2 - 481*n - 198)*x^2 - (18*n^2 + 180*n)*x^3)*v[i - 1] + (169 + (169*n + 429)*x + (13*n^2 + 416*n + 359)*x^2 + (22*n^2 + 337*n + 99)*x^3 + (9*n^2 + 90*n)*x^4)*deriv(v[i - 1], x))/(78*n - 78 + (39*n^2 + 33*n - 72)*x + (3*n^3 + 27*n^2 - 30*n)*x^2)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-5d39164c4108e043","steps":["basis A007318","\u03b8+1"],"edges":["e-0e2d8787d6ee9a58","e-958989a020502bff"],"text":"A099097 \u2192 basis A007318 \u2192 \u03b8+1"},"headline":"A099097 \u2192 basis A007318 \u2192 \u03b8+1","chain":null}