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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":"53f15e67d757dfe8603f44d1e4b9f923044ba2b9782a79cc4fd5fd6186712e47","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-180398bbec0732d0","role":"sequence","name":"shift_x_plus_one of Triangle read by rows: coefficients d(n,k) of Andr\u00e9 polynomials D(x,n) = Sum_{k>0} d(n,k)*x^k.","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:     2,      2;\n  3:     5,      8;\n  4:    16,     38,     12;\n  5:    61,    188,    102;\n  6:   272,   1038,    846,    136;\n  7:  1385,   6288,   6768,   1984;\n  8:  7936,  41806,  56232,  25088,  2480;\n  9: 50521, 302836, 488502, 291968, 55280;\n","recurrence_sentence":"The row polynomials satisfy -(x-n*x^3)*P(n,x) = (2*n-2+(n^2+4*n-8)*x-6*x^2-(n^3+4*n^2-2*n-8)*x^3-(2*n^3+2*n^2-8*n-8)*x^4-(n^3-n^2-2*n)*x^5)*P(n-2,x) - (1+(n+5)*x+(3*n+7)*x^2-(n^2+n+1)*x^3-(4*n^2+10*n+8)*x^4-(5*n^2+9*n+4)*x^5-(2*n^2+2*n)*x^6)*d/dx P(n-2,x) - (2+(2*n+2)*x-4*x^2-(2*n^2+3*n+2)*x^3-(2*n^2-2*n)*x^4)*P(n-1,x) + (x+3*x^2-(n-2)*x^3-3*n*x^4-2*n*x^5)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n - 2 + (n^2 + 4*n - 8)*x - 6*x^2 - (n^3 + 4*n^2 - 2*n - 8)*x^3 - (2*n^3 + 2*n^2 - 8*n - 8)*x^4 - (n^3 - n^2 - 2*n)*x^5)*P[n - 2, x] - (1 + (n + 5)*x + (3*n + 7)*x^2 - (n^2 + n + 1)*x^3 - (4*n^2 + 10*n + 8)*x^4 - (5*n^2 + 9*n + 4)*x^5 - (2*n^2 + 2*n)*x^6)*D[P[n - 2, x], x] - (2 + (2*n + 2)*x - 4*x^2 - (2*n^2 + 3*n + 2)*x^3 - (2*n^2 - 2*n)*x^4)*P[n - 1, x] + (x + 3*x^2 - (n - 2)*x^3 - 3*n*x^4 - 2*n*x^5)*D[P[n - 1, x], x])/(-x + n*x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((2*n - 2 + (n^2 + 4*n - 8)*x - 6*x^2 - (n^3 + 4*n^2 - 2*n - 8)*x^3 - (2*n^3 + 2*n^2 - 8*n - 8)*x^4 - (n^3 - n^2 - 2*n)*x^5)*v[i - 2] - (1 + (n + 5)*x + (3*n + 7)*x^2 - (n^2 + n + 1)*x^3 - (4*n^2 + 10*n + 8)*x^4 - (5*n^2 + 9*n + 4)*x^5 - (2*n^2 + 2*n)*x^6)*deriv(v[i - 2], x) - (2 + (2*n + 2)*x - 4*x^2 - (2*n^2 + 3*n + 2)*x^3 - (2*n^2 - 2*n)*x^4)*v[i - 1] + (x + 3*x^2 - (n - 2)*x^3 - 3*n*x^4 - 2*n*x^5)*deriv(v[i - 1], x))/(-x + n*x^3)); 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-ff270bcb802a85af","steps":["\u03b3 (scale 1/2) (shift +1)","x \u21a6 x+1","\u03b8+1"],"edges":["e-977fc5754e9861fd","e-6d784857f6230dda","e-4bfb0b5e34f58f7d"],"text":"A008292 \u2192 \u03b3 (scale 1/2) (shift +1) \u2192 x \u21a6 x+1 \u2192 \u03b8+1"},"headline":"A008292 \u2192 \u03b3 (scale 1/2) (shift +1) \u2192 x \u21a6 x+1 \u2192 \u03b8+1","chain":null}