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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":"17fb6023547faba61333b1964880d6b5553f65de8aa44ee10af089ef0e6a0337","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-1abe254a355aa2af","role":"sequence","name":"basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^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:           169,            312,            108;\n  1:          2197,           6084,           4212,             864;\n  2:         28561,         105456,         109512,           44928,            6480;\n  3:        371293,        1713660,        2372760,         1460160,          421200,           46656;\n  4:       4826809,       26733096,       46268820,        37964160,        16426800,         3639168,         326592;\n  5:      62748517,      405451956,      842092524,       863684640,       498279600,       165582144,       29719872,        2239488;\n  6:     815730721,     6023857632,    14596270416,     17964640512,     12955269600,      5740180992,     1545433344,      232906752,      15116544;\n  7:   10604499373,    88098917868,   243966234096,    350310489984,    303153308640,    167900294016,    60271900416,    13625044992,    1768635648,    100776960;\n  8:  137858491849,  1272539924760,  3964451304060,   6505766242560,   6568321687200,   4365407644416,  1958836763520,   590418616320,  114961317120,  13101004800,   665127936;\n  9: 1792160394037, 18197320924068, 62990726275620, 116290571585760, 134181428752800, 104042215525248, 56022731436672, 21107465533440, 5479822782720, 936721843200, 95113294848, 4353564672;\n","recurrence_sentence":"The row polynomials satisfy (13+(6*n+12)*x)*P(n,x) = (169+(78*n+312)*x+(36*n+108)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 169 + 312*x + 108*x^2.","mathematica":"P[0, x_] := 169 + 312*x + 108*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((169 + (78*n + 312)*x + (36*n + 108)*x^2)*P[n - 1, x])/(13 + (6*n + 12)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 169 + 312*x + 108*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((169 + (78*n + 312)*x + (36*n + 108)*x^2)*v[i - 1])/(13 + (6*n + 12)*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-4397e742863fdbe2","steps":["basis A013609","\u03b8+1"],"edges":["e-54bb49968ded72d0","e-29d9b45829c59742"],"text":"A038305 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A038305 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}