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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":"3338b1a94d3412b50069bb471b1a1ac8838cec65e10730bf621386ef8261ad9d","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-9a6b0e2632270c41","role":"sequence","name":"basis_map_A008277 of Riordan array (1-x,x/(1-x)).","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,     12,       9;\n  1:   2,     40,      63,       16;\n  2:   4,    128,     342,      192,        25;\n  3:   8,    400,    1656,     1568,       450,        36;\n  4:  16,   1232,    7524,    10784,      5150,       900,       49;\n  5:  32,   3760,   32868,    67296,     47750,     13716,     1617,       64;\n  6:  64,  11408,  139932,   394912,    391650,    164772,    31605,     2688,      81;\n  7: 128,  34480,  585396,  2223328,   2966950,   1717380,   477113,    65472,    4212,    100;\n  8: 256, 103952, 2419164, 12157344,  21275650,  16294068,  6154449,  1212416,  124983,   6300,  121;\n  9: 512, 312880, 9910548, 65087456, 146649750, 144695412, 71413629, 18950208, 2784294, 223600, 9075, 144;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(2+8*x-9*x^2)*P(n-2,x) - (4*x-8*x^2-4*x^3)*d/dx P(n-2,x) - (x^2-4*x^3)*d^2/dx^2 P(n-2,x) + (3+4*x)*P(n-1,x) + (2*x-4*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1 + 12*x + 9*x^2, P(1,x) = 2 + 40*x + 63*x^2 + 16*x^3.","mathematica":"P[0, x_] := 1 + 12*x + 9*x^2;\nP[1, x_] := 2 + 40*x + 63*x^2 + 16*x^3;\nP[n_, x_] := P[n, x] = Expand[-(2 + 8*x - 9*x^2)*P[n - 2, x] - (4*x - 8*x^2 - 4*x^3)*D[P[n - 2, x], x] - (x^2 - 4*x^3)*D[P[n - 2, x], {x, 2}] + (3 + 4*x)*P[n - 1, x] + (2*x - 4*x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1 + 12*x + 9*x^2; v[2] = 2 + 40*x + 63*x^2 + 16*x^3; for(n = 2, N, my(i = n + 1); v[i] = -(2 + 8*x - 9*x^2)*v[i - 2] - (4*x - 8*x^2 - 4*x^3)*deriv(v[i - 2], x) - (x^2 - 4*x^3)*deriv(deriv(v[i - 2], x), x) + (3 + 4*x)*v[i - 1] + (2*x - 4*x^2)*deriv(v[i - 1], x)); 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-7717563017cb9e5f","steps":["basis A008277","(\u03b8+1)\u00b2"],"edges":["e-94a89e959b0863ff","e-5ba65aa9603beda7"],"text":"A159854 \u2192 basis A008277 \u2192 (\u03b8+1)\u00b2"},"headline":"A159854 \u2192 basis A008277 \u2192 (\u03b8+1)\u00b2","chain":null}