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f(k) x^k maps every real-rooted polynomial to a real-rooted polynomial or zero (Laguerre; see T. 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":"9a5659acc26c1bb3f8058534139e203ddcf982add5b31167868a34e0d56386bd","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-0026e6b972fa04d4","role":"sequence","name":"Stirling2 triangle with scaled diagonals (powers of 9).","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:          81,             54,               3;\n  1:         729,           1134,             162,               4;\n  2:        6561,          21870,            6075,             360,              5;\n  3:       59049,         406782,          196830,           21060,            675,             6;\n  4:      531441,        7440174,         5924583,         1020600,          56700,          1134,            7;\n  5:     4782969,      134986014,       171124002,        44641044,        3827250,        129276,         1764,          8;\n  6:    43046721,     2439314190,      4822827075,      1835242920,      228027555,      11573604,       261954,       2592,         9;\n  7:   387420489,    43993748862,    133875302310,     72499181220,    12555293625,     898607682,     30005640,     486000,      3645,      10;\n  8:  3486784401,   792662320494,   3680623785663,   2788470927000,   655612189650,   63591167178,   2938730949,   69284160,    841995,    4950,   11;\n  9: 31381059609, 14274895337694, 100565835693642, 105292451752884, 32988137193000, 4220657655192, 259329744828, 8347009176, 146146275, 1381050, 6534, 12;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -x^2*P(n-2,x) - 9*x^2*d/dx P(n-2,x) + (9+2*x)*P(n-1,x) + 9*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 81 + 54*x + 3*x^2, P(1,x) = 729 + 1134*x + 162*x^2 + 4*x^3.","mathematica":"P[0, x_] := 81 + 54*x + 3*x^2;\nP[1, x_] := 729 + 1134*x + 162*x^2 + 4*x^3;\nP[n_, x_] := P[n, x] = Expand[-x^2*P[n - 2, x] - 9*x^2*D[P[n - 2, x], x] + (9 + 2*x)*P[n - 1, x] + 9*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] = 81 + 54*x + 3*x^2; v[2] = 729 + 1134*x + 162*x^2 + 4*x^3; for(n = 2, N, my(i = n + 1); v[i] = -x^2*v[i - 2] - 9*x^2*deriv(v[i - 2], x) + (9 + 2*x)*v[i - 1] + 9*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-0026e6b972fa04d4","steps":["\u03b8+1"],"edges":["e-06eead754dcd9d1a"],"text":"A075504 \u2192 \u03b8+1"},"headline":"A075504 \u2192 \u03b8+1","chain":null}