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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":"274db9643a39ba36071d5bc9a9348543d3a57531a71e08ad834279f99c4f3d87","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-267d61095455b8c8","role":"sequence","name":"reversal of Triangle read by rows: T(1,1)=1,T(2,1)=1,T(2,2)=3, T(k-1,r-1)+T(k-1,r)+T(k-2,r-1).","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: 3,  2;\n  2: 3, 10,   3;\n  3: 3, 22,  21,    4;\n  4: 3, 34,  69,   36,    5;\n  5: 3, 46, 153,  156,   55,    6;\n  6: 3, 58, 273,  452,  295,   78,    7;\n  7: 3, 70, 429, 1020, 1055,  498,  105,    8;\n  8: 3, 82, 621, 1956, 2895, 2118,  777,  136,   9;\n  9: 3, 94, 849, 3356, 6615, 6858, 3829, 1144, 171, 10;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -x^2*P(n-4,x) - (2*x+2*x^2)*P(n-3,x) - (1+x^2)*P(n-2,x) + (2+2*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 3 + 2*x, P(2,x) = 3 + 10*x + 3*x^2, P(3,x) = 3 + 22*x + 21*x^2 + 4*x^3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 2*x;\nP[2, x_] := 3 + 10*x + 3*x^2;\nP[3, x_] := 3 + 22*x + 21*x^2 + 4*x^3;\nP[n_, x_] := P[n, x] = Expand[-x^2*P[n - 4, x] - (2*x + 2*x^2)*P[n - 3, x] - (1 + x^2)*P[n - 2, x] + (2 + 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] = 3 + 2*x; v[3] = 3 + 10*x + 3*x^2; v[4] = 3 + 22*x + 21*x^2 + 4*x^3; for(n = 4, N, my(i = n + 1); v[i] = -x^2*v[i - 4] - (2*x + 2*x^2)*v[i - 3] - (1 + x^2)*v[i - 2] + (2 + 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-14dfd91463cae288","steps":["reverse","\u03b8+1"],"edges":["e-21b50016cc9ebe72","e-72703da55b2cb92a"],"text":"A102662 \u2192 reverse \u2192 \u03b8+1"},"headline":"A102662 \u2192 reverse \u2192 \u03b8+1","chain":null}