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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":"cbd06edbd4f3f3f4287fc8df83006b9df2a920743999936f76adccaae852ed0f","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-87eef110d487e3a4","role":"sequence","name":"basis_map_A007318 of Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).","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:     16,      24,       3;\n  3:     64,     160,      72,       4;\n  4:    256,     896,     720,     160,       5;\n  5:   1024,    4608,    5376,    2240,     300,       6;\n  6:   4096,   22528,   34560,   21504,    5600,     504,      7;\n  7:  16384,  106496,  202752,  168960,   67200,   12096,    784,     8;\n  8:  65536,  491520, 1118208, 1171456,  633600,  177408,  23520,  1152,    9;\n  9: 262144, 2228224, 5898240, 7454720, 5125120, 1976832, 413952, 42240, 1620, 10;\n","recurrence_sentence":"The row polynomials satisfy (8*n^2-4*n+(4*n^3-4*n^2+n)*x)*P(n,x) = (32*n^2-16*n+(16*n^3+32*n^2-36*n)*x+(12*n^3-13*n+3)*x^2)*P(n-1,x) - (16*n*x+(8*n^2+24*n-2)*x^2+(8*n^2+8*n-2)*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((32*n^2 - 16*n + (16*n^3 + 32*n^2 - 36*n)*x + (12*n^3 - 13*n + 3)*x^2)*P[n - 1, x] - (16*n*x + (8*n^2 + 24*n - 2)*x^2 + (8*n^2 + 8*n - 2)*x^3)*D[P[n - 1, x], x])/(8*n^2 - 4*n + (4*n^3 - 4*n^2 + n)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((32*n^2 - 16*n + (16*n^3 + 32*n^2 - 36*n)*x + (12*n^3 - 13*n + 3)*x^2)*v[i - 1] - (16*n*x + (8*n^2 + 24*n - 2)*x^2 + (8*n^2 + 8*n - 2)*x^3)*deriv(v[i - 1], x))/(8*n^2 - 4*n + (4*n^3 - 4*n^2 + n)*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-c30cb06ad75c5e21","steps":["x \u21a6 x+1","\u03b8+1"],"edges":["e-67136cef6b741eb8","e-b6bf6e7c83bd0096"],"text":"A103327 \u2192 x \u21a6 x+1 \u2192 \u03b8+1"},"headline":"A103327 \u2192 x \u21a6 x+1 \u2192 \u03b8+1","chain":null}