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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":"8929bab35ac0f2689ad05c153c2dee8f172672127517ef9edac7e4b76a006580","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,               2;\n  2: 0,              12,                9;\n  3: 0,             240,              270,               60;\n  4: 0,           10080,            13860,             5040,             525;\n  5: 0,          725760,          1134000,           529200,           94500,           5670;\n  6: 0,        79833600,        136715040,         74844000,        17671500,        1871100,         72765;\n  7: 0,     12454041600,      22884301440,      14045391360,      3972969000,      567567000,      39729690,      1081080;\n  8: 0,   2615348736000,    5085919238400,    3407218214400,   1097674578000,   190702512000,   18275657400,    908107200,    18243225;\n  9: 0, 711374856192000, 1450061425612800, 1042044190387200, 370970872272000, 74263644655200, 8753245300800, 602075073600, 22329707400, 344594250;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(16*n^4-96*n^3+204*n^2-176*n+48+(16*n^3-64*n^2+76*n-24)*x+(4*n^2-8*n+3)*x^2)*P(n-2,x) + (8*n^2-16*n+6+(4*n-2)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2*x;\nP[n_, x_] := P[n, x] = Expand[-(16*n^4 - 96*n^3 + 204*n^2 - 176*n + 48 + (16*n^3 - 64*n^2 + 76*n - 24)*x + (4*n^2 - 8*n + 3)*x^2)*P[n - 2, x] + (8*n^2 - 16*n + 6 + (4*n - 2)*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(16*n^4 - 96*n^3 + 204*n^2 - 176*n + 48 + (16*n^3 - 64*n^2 + 76*n - 24)*x + (4*n^2 - 8*n + 3)*x^2)*v[i - 2] + (8*n^2 - 16*n + 6 + (4*n - 2)*x)*v[i - 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-1336d058dca9979e","steps":["\u03b8+1"],"edges":["e-e5758d7f0957872d"],"text":"A204420 \u2192 \u03b8+1"},"headline":"A204420 \u2192 \u03b8+1","chain":null}