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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":"73b9df34004818b2e9f4e2a6659d6952903a254ee8592beeb0afc6a8e7b5b360","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-e88ccd05662b8cd9","role":"sequence","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*1^j.","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:          13,           8;\n  2:         169,         208,          36;\n  3:        2197,        4056,        1404,         128;\n  4:       28561,       70304,       36504,        6656,         400;\n  5:      371293,     1142440,      790920,      216320,       26000,       1152;\n  6:     4826809,    17822064,    15422940,     5624320,     1014000,      89856,      3136;\n  7:    62748517,   270301304,   280697508,   127953280,    30758000,    4088448,    285376,     8192;\n  8:   815730721,  4015905088,  4865423472,  2661428224,   799708000,  141732864,  14839552,   851968,   20736;\n  9: 10604499373, 58732611912, 81322078032, 51897850368, 18713167200, 4145686272, 578742528, 49840128, 2426112, 51200;\n","recurrence_sentence":"The row polynomials satisfy (169+(78*n-26)*x+4*n^2*x^2)*P(n,x) = (2197+(1014*n+1014)*x+(52*n^2+260*n+156)*x^2+(8*n^2+16*n+8)*x^3)*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[((2197 + (1014*n + 1014)*x + (52*n^2 + 260*n + 156)*x^2 + (8*n^2 + 16*n + 8)*x^3)*P[n - 1, x])/(169 + (78*n - 26)*x + 4*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] = ((2197 + (1014*n + 1014)*x + (52*n^2 + 260*n + 156)*x^2 + (8*n^2 + 16*n + 8)*x^3)*v[i - 1])/(169 + (78*n - 26)*x + 4*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-81e6fb4f9f226d77","steps":["basis A013609","(\u03b8+1)\u00b2"],"edges":["e-ec56a67a9fb88114","e-4b2c3f70e0defc80"],"text":"A038327 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2"},"headline":"A038327 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2","chain":null}