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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":"5e5fb6e8fc3dce81c54de3f24f7adf222c3e84f209a2a41dca980d3f56c7fc06","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-02156bdfcfec6826","role":"sequence","name":"Triangle of coefficients of p(n, x) where p(n, x) is defined as p(n, x) = (1-3*x)^(n+1)*PolyLog(-n, 3*x)/(3*x), read by rows.","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,    24,       27;\n  1: 1,    66,      297,        108;\n  2: 1,   156,     1782,       2808,         405;\n  3: 1,   342,     8154,      32616,       23085,         1458;\n  4: 1,   720,    32157,     260928,      482355,       174960,         5103;\n  5: 1,  1482,   115911,    1686852,     6325695,      6259194,      1260441,         17496;\n  6: 1,  3012,   394416,    9529272,    63256950,    128645172,     74544624,       8782992,        59049;\n  7: 1,  6078,  1291680,   49160736,   530693370,   1910496132,   2322844776,     837008640,     59816637,      196830;\n  8: 1, 12216,  4121199,  237976704,  3943936170,  22925953584,  49693595742,   38552226048,   9013062213,   400745880,     649539;\n  9: 1, 24498, 12913317, 1100269980, 26858981970, 236942912988, 829300195458, 1160308021104, 601572611565, 94138080930, 2652067737, 2125764;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(9*n^2+36*n+27)*x^2*P(n-2,x) - ((3*n+9)*x^2-(9*n+27)*x^3)*d/dx P(n-2,x) + (1+(6*n+12)*x)*P(n-1,x) + (x-3*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1 + 24*x + 27*x^2, P(1,x) = 1 + 66*x + 297*x^2 + 108*x^3.","mathematica":"P[0, x_] := 1 + 24*x + 27*x^2;\nP[1, x_] := 1 + 66*x + 297*x^2 + 108*x^3;\nP[n_, x_] := P[n, x] = Expand[-(9*n^2 + 36*n + 27)*x^2*P[n - 2, x] - ((3*n + 9)*x^2 - (9*n + 27)*x^3)*D[P[n - 2, x], x] + (1 + (6*n + 12)*x)*P[n - 1, x] + (x - 3*x^2)*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] = 1 + 24*x + 27*x^2; v[2] = 1 + 66*x + 297*x^2 + 108*x^3; for(n = 2, N, my(i = n + 1); v[i] = -(9*n^2 + 36*n + 27)*x^2*v[i - 2] - ((3*n + 9)*x^2 - (9*n + 27)*x^3)*deriv(v[i - 2], x) + (1 + (6*n + 12)*x)*v[i - 1] + (x - 3*x^2)*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-02156bdfcfec6826","steps":["\u03b8+1"],"edges":["e-7e0e37d5fc982e34"],"text":"A156366 \u2192 \u03b8+1"},"headline":"A156366 \u2192 \u03b8+1","chain":null}