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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":"40207130e5faf28440deb292bece73ed052778133873d1aae7afb6cb0af6b38b","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-5b7da9f56b3380cb","role":"sequence","name":"Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 7*x + 2.","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,         4;\n  2:   4,        72,          12;\n  3:   8,       776,        1164,            32;\n  4:  16,      7352,       37248,         14704,            80;\n  5:  32,     67128,      849612,       1132816,        167820,           192;\n  6:  64,    606520,    16614552,      52109536,      27690920,       1819560,           448;\n  7: 128,   5464312,   299494692,    1863101408,    2328876760,     598989384,      19125092,         1024;\n  8: 256,  49191864,  5152559664,   57626403856,  139732605600,   86439605784,   12022639216,    196767456,       2304;\n  9: 512, 442756472, 86204132220, 1627690788976, 6857199346340, 8228639215608, 2848458880708, 229877685920, 1992404124, 5120;\n","recurrence_sentence":"The row polynomials satisfy (7*n-5)*P(n,x) = (70*n+20+(490*n^2-700*n-240)*x-(343*n^3-735*n^2+182*n+120)*x^2)*P(n-2,x) + ((245*n+70)*x-(343*n^2+98*n)*x^2+(343*n^2-147*n-70)*x^3)*d/dx P(n-2,x) - (21*n+20-(98*n^2-140*n+50)*x)*P(n-1,x) + ((49*n-35)*x-(49*n-35)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[((70*n + 20 + (490*n^2 - 700*n - 240)*x - (343*n^3 - 735*n^2 + 182*n + 120)*x^2)*P[n - 2, x] + ((245*n + 70)*x - (343*n^2 + 98*n)*x^2 + (343*n^2 - 147*n - 70)*x^3)*D[P[n - 2, x], x] - (21*n + 20 - (98*n^2 - 140*n + 50)*x)*P[n - 1, x] + ((49*n - 35)*x - (49*n - 35)*x^2)*D[P[n - 1, x], x])/(7*n - 5)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((70*n + 20 + (490*n^2 - 700*n - 240)*x - (343*n^3 - 735*n^2 + 182*n + 120)*x^2)*v[i - 2] + ((245*n + 70)*x - (343*n^2 + 98*n)*x^2 + (343*n^2 - 147*n - 70)*x^3)*deriv(v[i - 2], x) - (21*n + 20 - (98*n^2 - 140*n + 50)*x)*v[i - 1] + ((49*n - 35)*x - (49*n - 35)*x^2)*deriv(v[i - 1], x))/(7*n - 5)); 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-5b7da9f56b3380cb","steps":["\u03b8+1"],"edges":["e-db5a41ff6346fafc"],"text":"A257617 \u2192 \u03b8+1"},"headline":"A257617 \u2192 \u03b8+1","chain":null}