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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":"be6e8c913614da5b13642860deb118150fb36b645632bef74d7b0f3fd3b79ec8","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-3fea9ad1525b48eb","role":"sequence","name":"Triangle of Ward numbers T(n,k) read by rows.","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: 1,   12;\n   3: 1,   40,     135;\n   4: 1,  100,     945,      1680;\n   5: 1,  224,    4410,     20160,      23625;\n   6: 1,  476,   17262,    151200,     433125,      374220;\n   7: 1,  984,   61425,    911680,    4764375,     9729720,     6621615;\n   8: 1, 2004,  206415,   4847920,   40915875,   147567420,   231756525,    129729600;\n   9: 1, 4048,  668844,  23795200,  303052750,  1710268560,  4635130500,   5881075200,   2791213425;\n  10: 1, 8140, 2115828, 110638528, 2035783750, 16807550760, 69691762140, 149967417600, 159099165225, 65472907500;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = -(4*n^2*x-(9*n^3-7*n^2-2*n)*x^2)*P(n-2,x) - (2*n*x-(4*n^2-8*n-1)*x^2-(4*n^3+10*n^2-11*n-3)*x^3)*d/dx P(n-2,x) - (n*x^2-(4*n^2-4*n-1)*x^3-(4*n^2-3*n-1)*x^4)*d^2/dx^2 P(n-2,x) + (n-1+(4*n^2-4)*x)*P(n-1,x) + ((2*n-1)*x-(4*n^2-4*n)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 12*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1 + 12*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(4*n^2*x - (9*n^3 - 7*n^2 - 2*n)*x^2)*P[n - 2, x] - (2*n*x - (4*n^2 - 8*n - 1)*x^2 - (4*n^3 + 10*n^2 - 11*n - 3)*x^3)*D[P[n - 2, x], x] - (n*x^2 - (4*n^2 - 4*n - 1)*x^3 - (4*n^2 - 3*n - 1)*x^4)*D[P[n - 2, x], {x, 2}] + (n - 1 + (4*n^2 - 4)*x)*P[n - 1, x] + ((2*n - 1)*x - (4*n^2 - 4*n)*x^2)*D[P[n - 1, x], x])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 12*x; for(n = 3, N, my(i = n); v[i] = (-(4*n^2*x - (9*n^3 - 7*n^2 - 2*n)*x^2)*v[i - 2] - (2*n*x - (4*n^2 - 8*n - 1)*x^2 - (4*n^3 + 10*n^2 - 11*n - 3)*x^3)*deriv(v[i - 2], x) - (n*x^2 - (4*n^2 - 4*n - 1)*x^3 - (4*n^2 - 3*n - 1)*x^4)*deriv(deriv(v[i - 2], x), x) + (n - 1 + (4*n^2 - 4)*x)*v[i - 1] + ((2*n - 1)*x - (4*n^2 - 4*n)*x^2)*deriv(v[i - 1], x))/(n - 1)); 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-3fea9ad1525b48eb","steps":["(\u03b8+1)\u00b2"],"edges":["e-b2ea7b6e80a7790c"],"text":"A134991 \u2192 (\u03b8+1)\u00b2"},"headline":"A134991 \u2192 (\u03b8+1)\u00b2","chain":null}