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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":"329b1a654599f638161fb93da55398df8a57a33b394a9a6c3f61b374c5ee8ea3","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-1336d058dca9979e","role":"sequence","name":"Triangle T(n,k) giving number of degree-2n permutations which decompose into exactly k cycles of even length, k=0..n.","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: 0,                4;\n  2: 0,               24,               27;\n  3: 0,              480,              810,              240;\n  4: 0,            20160,            41580,            20160,             2625;\n  5: 0,          1451520,          3402000,          2116800,           472500,           34020;\n  6: 0,        159667200,        410145120,        299376000,         88357500,        11226600,         509355;\n  7: 0,      24908083200,      68652904320,      56181565440,      19864845000,      3405402000,      278107830,       8648640;\n  8: 0,    5230697472000,   15257757715200,   13628872857600,    5488372890000,   1144215072000,   127929601800,    7264857600,    164189025;\n  9: 0, 1422749712384000, 4350184276838400, 4168176761548800, 1854854361360000, 445581867931200, 61272717105600, 4816600588800, 200967366600, 3445942500;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(16*n^4-96*n^3+204*n^2-176*n+48+(32*n^3-128*n^2+152*n-48)*x-(36*n^2-72*n+27)*x^2)*P(n-2,x) + ((32*n^3-128*n^2+152*n-48)*x^2+(16*n^2-32*n+12)*x^3)*d/dx P(n-2,x) + (8*n^2-16*n+6+(8*n-4)*x)*P(n-1,x) - (8*n-4)*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4*x;\nP[n_, x_] := P[n, x] = Expand[-(16*n^4 - 96*n^3 + 204*n^2 - 176*n + 48 + (32*n^3 - 128*n^2 + 152*n - 48)*x - (36*n^2 - 72*n + 27)*x^2)*P[n - 2, x] + ((32*n^3 - 128*n^2 + 152*n - 48)*x^2 + (16*n^2 - 32*n + 12)*x^3)*D[P[n - 2, x], x] + (8*n^2 - 16*n + 6 + (8*n - 4)*x)*P[n - 1, x] - (8*n - 4)*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; v[2] = 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(16*n^4 - 96*n^3 + 204*n^2 - 176*n + 48 + (32*n^3 - 128*n^2 + 152*n - 48)*x - (36*n^2 - 72*n + 27)*x^2)*v[i - 2] + ((32*n^3 - 128*n^2 + 152*n - 48)*x^2 + (16*n^2 - 32*n + 12)*x^3)*deriv(v[i - 2], x) + (8*n^2 - 16*n + 6 + (8*n - 4)*x)*v[i - 1] - (8*n - 4)*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-1336d058dca9979e","steps":["(\u03b8+1)\u00b2"],"edges":["e-058e09cdedf2d961"],"text":"A204420 \u2192 (\u03b8+1)\u00b2"},"headline":"A204420 \u2192 (\u03b8+1)\u00b2","chain":null}