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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":"a7c50602879b7c0b498830c6f7da06928c559e8a65eae1e587c1efe52c97ed0a","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-f00f6de2d4bbc6f5","role":"sequence","name":"Triangle read by rows: the reversed x = 1+q Narayana triangle at m=2.","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:       3,       2;\n   3:      12,      16,        3;\n   4:      55,     110,       45,        4;\n   5:     273,     728,      468,       96,       5;\n   6:    1428,    4760,     4200,     1400,     175,       6;\n   7:    7752,   31008,    34884,    16320,    3400,     288,      7;\n   8:   43263,  201894,   276507,   167580,   49875,    7182,    441,     8;\n   9:  246675, 1315600,  2125200,  1586816,  619850,  129360,  13720,   640,   9;\n  10: 1430715, 8584290, 15984540, 14208480, 6906900, 1912680, 297528, 24288, 891, 10;\n","recurrence_sentence":"The row polynomials satisfy (12*n^3+2*n^2-2*n+(4*n^4-2*n^3-2*n^2)*x)*P(n,x) = (81*n^3-108*n^2+45*n-6+(27*n^4+18*n^3-87*n^2+50*n-8)*x+(12*n^4-12*n^3-10*n^2+14*n-4)*x^2)*P(n-1,x) - ((27*n^2-18*n+3)*x+(9*n^3+24*n^2-19*n+4)*x^2+(8*n^3+2*n^2-6*n+2)*x^3)*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[((81*n^3 - 108*n^2 + 45*n - 6 + (27*n^4 + 18*n^3 - 87*n^2 + 50*n - 8)*x + (12*n^4 - 12*n^3 - 10*n^2 + 14*n - 4)*x^2)*P[n - 1, x] - ((27*n^2 - 18*n + 3)*x + (9*n^3 + 24*n^2 - 19*n + 4)*x^2 + (8*n^3 + 2*n^2 - 6*n + 2)*x^3)*D[P[n - 1, x], x])/(12*n^3 + 2*n^2 - 2*n + (4*n^4 - 2*n^3 - 2*n^2)*x)]];\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] = ((81*n^3 - 108*n^2 + 45*n - 6 + (27*n^4 + 18*n^3 - 87*n^2 + 50*n - 8)*x + (12*n^4 - 12*n^3 - 10*n^2 + 14*n - 4)*x^2)*v[i - 1] - ((27*n^2 - 18*n + 3)*x + (9*n^3 + 24*n^2 - 19*n + 4)*x^2 + (8*n^3 + 2*n^2 - 6*n + 2)*x^3)*deriv(v[i - 1], x))/(12*n^3 + 2*n^2 - 2*n + (4*n^4 - 2*n^3 - 2*n^2)*x)); 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-f00f6de2d4bbc6f5","steps":["\u03b8+1"],"edges":["e-26a22938063ab36f"],"text":"A243662 \u2192 \u03b8+1"},"headline":"A243662 \u2192 \u03b8+1","chain":null}