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Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. 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":"39033a5739d24c30aa9e8845e4405f3fad7cdbfbf89c2f34b6282df7768ad9be","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-5c56442adc0376c2","role":"sequence","name":"Triangle read by rows, the Bell transform of the quartic factorial numbers A007696(n+1) without column 0.","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:            5,            2;\n   3:           45,           30,            3;\n   4:          585,          510,           90,           4;\n   5:         9945,        10350,         2475,         200,          5;\n   6:       208845,       247590,        72450,        8100,        375,         6;\n   7:      5221125,      6855030,      2327535,      323400,      21000,       630,       7;\n   8:    151412625,    216093150,     82436130,    13452180,    1097250,     46620,     980,      8;\n   9:   4996616625,   7649992350,   3209404275,   594193320,   57413475,   3088260,   92610,   1440,    9;\n  10: 184874815125, 300692942550, 136641755250, 28046938500, 3096694125, 198603090, 7585200, 169200, 2025, 10;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(16*n^2-60*n+56+(8*n-16)*x+x^2)*P(n-2,x) - ((4*n-8)*x+x^2)*d/dx P(n-2,x) + (8*n-11+2*x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 5 + 2*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 5 + 2*x;\nP[n_, x_] := P[n, x] = Expand[-(16*n^2 - 60*n + 56 + (8*n - 16)*x + x^2)*P[n - 2, x] - ((4*n - 8)*x + x^2)*D[P[n - 2, x], x] + (8*n - 11 + 2*x)*P[n - 1, x] + x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 5 + 2*x; for(n = 3, N, my(i = n); v[i] = -(16*n^2 - 60*n + 56 + (8*n - 16)*x + x^2)*v[i - 2] - ((4*n - 8)*x + x^2)*deriv(v[i - 2], x) + (8*n - 11 + 2*x)*v[i - 1] + 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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-5c56442adc0376c2","steps":["\u03b8+1"],"edges":["e-d23510fcca210c5a"],"text":"A049029 \u2192 \u03b8+1"},"headline":"A049029 \u2192 \u03b8+1","chain":null}