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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":"2b0c85823cf1a62fe672f4de740570c094210d617adbe27ad3777c5ea6332373","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-9b40f045d129e237","role":"sequence","name":"Coefficients of a generalized Jaco-Lucas polynomial (even indices) read by rows.","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:    2;\n  1:    3,     4;\n  2:    7,    16,      6;\n  3:   18,    60,     45,      8;\n  4:   47,   208,    240,     96,     10;\n  5:  123,   680,   1065,    680,    175,     12;\n  6:  322,  2136,   4230,   3728,   1575,    288,    14;\n  7:  843,  6524,  15624,  17584,  10395,   3192,   441,    16;\n  8: 2207, 19520,  54840,  75136,  57200,  24864,  5880,   640,  18;\n  9: 5778, 57492, 185355, 299352, 279045, 157356, 53214, 10080, 891, 20;\n","recurrence_sentence":"The row polynomials satisfy (20*n-20+(20*n^2-18*n-2)*x+(8*n^3-12*n^2+4*n)*x^2)*P(n,x) = (80*n-80+(100*n^2-2*n-98)*x+(32*n^3+72*n^2-92*n-12)*x^2+(24*n^3-4*n^2-32*n+12)*x^3)*P(n-1,x) - (25+(50*n+50)*x+(20*n^2+110*n+19)*x^2+(36*n^2+76*n-14)*x^3+(16*n^2+16*n-8)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 2, P(1,x) = 3 + 4*x.","mathematica":"P[0, x_] := 2;\nP[1, x_] := 3 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[((80*n - 80 + (100*n^2 - 2*n - 98)*x + (32*n^3 + 72*n^2 - 92*n - 12)*x^2 + (24*n^3 - 4*n^2 - 32*n + 12)*x^3)*P[n - 1, x] - (25 + (50*n + 50)*x + (20*n^2 + 110*n + 19)*x^2 + (36*n^2 + 76*n - 14)*x^3 + (16*n^2 + 16*n - 8)*x^4)*D[P[n - 1, x], x])/(20*n - 20 + (20*n^2 - 18*n - 2)*x + (8*n^3 - 12*n^2 + 4*n)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 2; v[2] = 3 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((80*n - 80 + (100*n^2 - 2*n - 98)*x + (32*n^3 + 72*n^2 - 92*n - 12)*x^2 + (24*n^3 - 4*n^2 - 32*n + 12)*x^3)*v[i - 1] - (25 + (50*n + 50)*x + (20*n^2 + 110*n + 19)*x^2 + (36*n^2 + 76*n - 14)*x^3 + (16*n^2 + 16*n - 8)*x^4)*deriv(v[i - 1], x))/(20*n - 20 + (20*n^2 - 18*n - 2)*x + (8*n^3 - 12*n^2 + 4*n)*x^2)); 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-9b40f045d129e237","steps":["\u03b8+1"],"edges":["e-e30319ec85031216"],"text":"A122076 \u2192 \u03b8+1"},"headline":"A122076 \u2192 \u03b8+1","chain":null}