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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":"f0b521ace48dbd258b370b1d5a38422416efc9b8613ffd23d73545aee790f747","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-d4d62d9c5cf9eec8","role":"sequence","name":"basis_map_A013610 of Triangle read by rows: T(n,k) = |s(n,n+1-k)|, where s(n,k) are the signed Stirling numbers of the first kind A008276 (1 <= k <= n; in other words, the unsigned Stirling numbers of the first kind in reverse order).","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;\n  1:       2,         6;\n  2:       6,        42,         54;\n  3:      24,       276,        783,         648;\n  4:     120,      1956,       8883,       15768,        9720;\n  5:     720,     15336,      97308,      272268,      353970,       174960;\n  6:    5040,    133272,    1095228,     4241268,     8603820,      8870472,      3674160;\n  7:   40320,   1277856,   12959892,    64596528,   180163845,    287780040,    246719844,     88179840;\n  8:  362880,  13436064,  162641844,   996085296,  3559370445,   7778739096,  10278319716,   7560791424,   2380855680;\n  9: 3628800, 153956160, 2170579032, 15815959344, 69211583190, 193110993378, 347813478684, 392767494048, 253467596304, 71425670400;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = -(n^3+(6*n^3-6*n^2)*x+(9*n^3-18*n^2+9*n)*x^2)*P(n-2,x) + (2*n^2-1+(6*n^2-6*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 6*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 6*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + (6*n^3 - 6*n^2)*x + (9*n^3 - 18*n^2 + 9*n)*x^2)*P[n - 2, x] + (2*n^2 - 1 + (6*n^2 - 6*n)*x)*P[n - 1, x])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 + (6*n^3 - 6*n^2)*x + (9*n^3 - 18*n^2 + 9*n)*x^2)*v[i - 2] + (2*n^2 - 1 + (6*n^2 - 6*n)*x)*v[i - 1])/(n - 1)); 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-4d60c3dd8627c01d","steps":["basis A013610","\u03b8+1"],"edges":["e-af0dc46429e1a2ef","e-e296ce4f773eb01a"],"text":"A094638 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A094638 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}