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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":"06d75bc9e533db17811e610b520411f041927176dd1d518484afb9efd87678e7","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-61f790c1436a0d7c","role":"sequence","name":"Triangle T(n,k) of associated Stirling numbers of second kind, n >= 2, 1 <= k <= floor(n/2).","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: 1;\n  2: 1,   12;\n  3: 1,   40;\n  4: 1,  100,    135;\n  5: 1,  224,    945;\n  6: 1,  476,   4410,   1680;\n  7: 1,  984,  17262,  20160;\n  8: 1, 2004,  61425, 151200,  23625;\n  9: 1, 4048, 206415, 911680, 433125;\n","recurrence_sentence":"The row polynomials satisfy n*P(n,x) = (9*n^3-9*n)*x^2*P(n-4,x) + (4*n^3-4*n)*x^3*d/dx P(n-4,x) - (4*n^2+4*n)*x*P(n-3,x) + (4*n^2+4*n)*x^2*d/dx P(n-3,x) + (4*n^2+4*n)*x^3*d^2/dx^2 P(n-3,x) + (4*n^2+4*n)*x*P(n-2,x) - ((2*n+2)*x+(4*n^2+4*n)*x^2)*d/dx P(n-2,x) - (n+1)*x^2*d^2/dx^2 P(n-2,x) + n*P(n-1,x) + (2*n+1)*x*d/dx P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1, P(2,x) = 1 + 12*x, P(3,x) = 1 + 40*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[2, x_] := 1 + 12*x;\nP[3, x_] := 1 + 40*x;\nP[n_, x_] := P[n, x] = Expand[Together[((9*n^3 - 9*n)*x^2*P[n - 4, x] + (4*n^3 - 4*n)*x^3*D[P[n - 4, x], x] - (4*n^2 + 4*n)*x*P[n - 3, x] + (4*n^2 + 4*n)*x^2*D[P[n - 3, x], x] + (4*n^2 + 4*n)*x^3*D[P[n - 3, x], {x, 2}] + (4*n^2 + 4*n)*x*P[n - 2, x] - ((2*n + 2)*x + (4*n^2 + 4*n)*x^2)*D[P[n - 2, x], x] - (n + 1)*x^2*D[P[n - 2, x], {x, 2}] + n*P[n - 1, x] + (2*n + 1)*x*D[P[n - 1, x], x])/(n)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; v[3] = 1 + 12*x; v[4] = 1 + 40*x; for(n = 4, N, my(i = n + 1); v[i] = ((9*n^3 - 9*n)*x^2*v[i - 4] + (4*n^3 - 4*n)*x^3*deriv(v[i - 4], x) - (4*n^2 + 4*n)*x*v[i - 3] + (4*n^2 + 4*n)*x^2*deriv(v[i - 3], x) + (4*n^2 + 4*n)*x^3*deriv(deriv(v[i - 3], x), x) + (4*n^2 + 4*n)*x*v[i - 2] - ((2*n + 2)*x + (4*n^2 + 4*n)*x^2)*deriv(v[i - 2], x) - (n + 1)*x^2*deriv(deriv(v[i - 2], x), x) + n*v[i - 1] + (2*n + 1)*x*deriv(v[i - 1], x))/(n)); 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-61f790c1436a0d7c","steps":["(\u03b8+1)\u00b2"],"edges":["e-8bd64729d41873de"],"text":"A008299 \u2192 (\u03b8+1)\u00b2"},"headline":"A008299 \u2192 (\u03b8+1)\u00b2","chain":null}