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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":"de7055d9a7f4fa1121d1296b5efd612d94c9f5c892f4e85316b0c23a23c19630","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-e0a0f9191d8a0512","role":"sequence","name":"basis_map_A007318 of Triangle read by rows: coefficients of the polynomials given by KummerU(-n, 1 - n - x, 3).","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:       4,        2;\n  2:      17,       18,        3;\n  3:      78,      130,       45,        4;\n  4:     393,      904,      483,       88,        5;\n  5:    2208,     6458,     4680,     1300,      150,       6;\n  6:   13977,    48978,    44562,    16620,     2900,     234,      7;\n  7:  100026,   401490,   434847,   202216,    47075,    5712,    343,     8;\n  8:  806769,  3587904,  4449750,  2455152,   709695,  114408,  10290,   480,   9;\n  9: 7280604, 35032626, 48342528, 30541640, 10455480, 2087442, 248724, 17328, 648, 10;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(9*n^2-45*n+54)*P(n-4,x) + (6*n^2-3*n-18+(6*n-12)*x)*P(n-3,x) - (n^2+10*n-5+(2*n+4)*x+x^2)*P(n-2,x) + (2*n+5+2*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 4 + 2*x, P(2,x) = 17 + 18*x + 3*x^2, P(3,x) = 78 + 130*x + 45*x^2 + 4*x^3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4 + 2*x;\nP[2, x_] := 17 + 18*x + 3*x^2;\nP[3, x_] := 78 + 130*x + 45*x^2 + 4*x^3;\nP[n_, x_] := P[n, x] = Expand[-(9*n^2 - 45*n + 54)*P[n - 4, x] + (6*n^2 - 3*n - 18 + (6*n - 12)*x)*P[n - 3, x] - (n^2 + 10*n - 5 + (2*n + 4)*x + x^2)*P[n - 2, x] + (2*n + 5 + 2*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + 2*x; v[3] = 17 + 18*x + 3*x^2; v[4] = 78 + 130*x + 45*x^2 + 4*x^3; for(n = 4, N, my(i = n + 1); v[i] = -(9*n^2 - 45*n + 54)*v[i - 4] + (6*n^2 - 3*n - 18 + (6*n - 12)*x)*v[i - 3] - (n^2 + 10*n - 5 + (2*n + 4)*x + x^2)*v[i - 2] + (2*n + 5 + 2*x)*v[i - 1]); 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-ff69dadb0811f5e6","steps":["x \u21a6 x+1","\u03b8+1"],"edges":["e-29bcfb0ecbd3591f","e-4a78d88761eb260a"],"text":"A327997 \u2192 x \u21a6 x+1 \u2192 \u03b8+1"},"headline":"A327997 \u2192 x \u21a6 x+1 \u2192 \u03b8+1","chain":null}