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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":"252480936c75698644c977b7759183e783bcd0cdbec938ebbd55b2582aa597e5","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-67d789c805b2e2cd","role":"sequence","name":"basis_map_A007318 of Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:    1;\n   2:    4,     2;\n   3:   11,    14,      3;\n   4:   29,    62,     30,      4;\n   5:   76,   230,    180,     52,      5;\n   6:  199,   780,    855,    392,     80,     6;\n   7:  521,  2508,   3555,   2264,    725,   114,     7;\n   8: 1364,  7786,  13572,  11140,   4925,  1206,   154,    8;\n   9: 3571, 23578,  48840,  49252,  27975,  9414,  1862,  200,   9;\n  10: 9349, 70090, 168315, 201736, 140565, 60606, 16415, 2720, 252, 10;\n","recurrence_sentence":"The row polynomials satisfy -(20*n-30+(8*n-12)*x-(2*n^3-9*n^2+13*n-6)*x^2)*P(n,x) = -(30*n-20+(5*n^2+17*n-18)*x-(3*n^3-11*n^2+4*n+4)*x^2-(n^3-3*n^2+2*n)*x^3)*P(n-1,x) - (25+50*x-(5*n^2-10*n-29)*x^2-(6*n^2-12*n-4)*x^3-(n^2-2*n)*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(30*n - 20 + (5*n^2 + 17*n - 18)*x - (3*n^3 - 11*n^2 + 4*n + 4)*x^2 - (n^3 - 3*n^2 + 2*n)*x^3)*P[n - 1, x] - (25 + 50*x - (5*n^2 - 10*n - 29)*x^2 - (6*n^2 - 12*n - 4)*x^3 - (n^2 - 2*n)*x^4)*D[P[n - 1, x], x])/(-20*n + 30 - (8*n - 12)*x + (2*n^3 - 9*n^2 + 13*n - 6)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(30*n - 20 + (5*n^2 + 17*n - 18)*x - (3*n^3 - 11*n^2 + 4*n + 4)*x^2 - (n^3 - 3*n^2 + 2*n)*x^3)*v[i - 1] - (25 + 50*x - (5*n^2 - 10*n - 29)*x^2 - (6*n^2 - 12*n - 4)*x^3 - (n^2 - 2*n)*x^4)*deriv(v[i - 1], x))/(-20*n + 30 - (8*n - 12)*x + (2*n^3 - 9*n^2 + 13*n - 6)*x^2)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-318700a428d7b97b","steps":["basis A007318","\u03b8+1"],"edges":["e-0092e3a0a8732095","e-bb55875c4585a31d"],"text":"A111125 \u2192 basis A007318 \u2192 \u03b8+1"},"headline":"A111125 \u2192 basis A007318 \u2192 \u03b8+1","chain":null}