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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":"5fc71c77b1aa4a7d6721e78f1edaa1d50d980739f92eb1c62357a65ceeb6d1ce","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-261c6497576cc5f7","role":"sequence","name":"basis_map_A007318 of Triangle of coefficients in expansion of (1+x)*(1+3x)*(1+5x)*(1+7x)*...*(1+(2n-1)x).","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,          2;\n  2:         8,         20,           9;\n  3:        48,        200,         204,          60;\n  4:       384,       2272,        3732,        2384,         525;\n  5:      3840,      29632,       67992,       68624,       32070,        5670;\n  6:     46080,     440064,     1304832,     1820704,     1328420,      491364,       72765;\n  7:    645120,    7358976,    26848896,    48106944,    48184320,    27602448,     8471064,     1081080;\n  8:  10321920,  137097216,   595159296,  1306689024,  1672954320,  1308956928,   618579864,   162515520,   18243225;\n  9: 185794560, 2818695168, 14208846336, 37010679808, 57880319520, 57689492832, 37095416624, 14943402432, 3436487370, 344594250;\n","recurrence_sentence":"The row polynomials satisfy (2*n-3)*P(n,x) = -(8*n^3-20*n^2+16*n-4+(16*n^3-48*n^2+44*n-12)*x+(8*n^3-28*n^2+30*n-9)*x^2)*P(n-2,x) + (8*n^2-12*n+2+(8*n^2-16*n+6)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(8*n^3 - 20*n^2 + 16*n - 4 + (16*n^3 - 48*n^2 + 44*n - 12)*x + (8*n^3 - 28*n^2 + 30*n - 9)*x^2)*P[n - 2, x] + (8*n^2 - 12*n + 2 + (8*n^2 - 16*n + 6)*x)*P[n - 1, x])/(2*n - 3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(8*n^3 - 20*n^2 + 16*n - 4 + (16*n^3 - 48*n^2 + 44*n - 12)*x + (8*n^3 - 28*n^2 + 30*n - 9)*x^2)*v[i - 2] + (8*n^2 - 12*n + 2 + (8*n^2 - 16*n + 6)*x)*v[i - 1])/(2*n - 3)); 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-2b477f2aa77e3efc","steps":["basis A007318","\u03b8+1"],"edges":["e-07dfe9939e4be29b","e-51e3407ffcce380e"],"text":"A109692 \u2192 basis A007318 \u2192 \u03b8+1"},"headline":"A109692 \u2192 basis A007318 \u2192 \u03b8+1","chain":null}