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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":"8b5676ae24502bb40ca5d149305058fd2734290908bf7eb93ce80b2ded50b905","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-fc18ad0ea0731402","role":"sequence","name":"Triangle read by rows: Stirling2 triangle with scaled diagonals (powers of 7).","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:         49,            42,              3;\n  1:        343,           686,            126,              4;\n  2:       2401,         10290,           3675,            280,             5;\n  3:      16807,        148862,          92610,          12740,           525,            6;\n  4:     117649,       2117682,        2168103,         480200,         34300,          882,           7;\n  5:     823543,      29882846,       48706686,       16336404,       1800750,        78204,        1372,          8;\n  6:    5764801,     420006930,     1067664675,      522361560,      83446755,      5445468,      158466,       2016,        9;\n  7:   40353607,    5891626622,    23050968570,    16049676580,    3573588375,    328845762,    14117880,     294000,     2835,     10;\n  8:  282475249,   82563479922,   492907779903,   480125569000,  145137688850,  18099828054,  1075429509,   32598720,   509355,   3850,   11;\n  9: 1977326743, 1156453669406, 10474908597846, 14100726305204, 5679976071000, 934358004888, 73812512004, 3054590616, 68762925, 835450, 5082, 12;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -x^2*P(n-2,x) - 7*x^2*d/dx P(n-2,x) + (7+2*x)*P(n-1,x) + 7*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 49 + 42*x + 3*x^2, P(1,x) = 343 + 686*x + 126*x^2 + 4*x^3.","mathematica":"P[0, x_] := 49 + 42*x + 3*x^2;\nP[1, x_] := 343 + 686*x + 126*x^2 + 4*x^3;\nP[n_, x_] := P[n, x] = Expand[-x^2*P[n - 2, x] - 7*x^2*D[P[n - 2, x], x] + (7 + 2*x)*P[n - 1, x] + 7*x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 49 + 42*x + 3*x^2; v[2] = 343 + 686*x + 126*x^2 + 4*x^3; for(n = 2, N, my(i = n + 1); v[i] = -x^2*v[i - 2] - 7*x^2*deriv(v[i - 2], x) + (7 + 2*x)*v[i - 1] + 7*x*deriv(v[i - 1], x)); 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-fc18ad0ea0731402","steps":["\u03b8+1"],"edges":["e-58eb1d3dd1f1c7b6"],"text":"A075502 \u2192 \u03b8+1"},"headline":"A075502 \u2192 \u03b8+1","chain":null}