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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":"ea760abbfc7465f0d2c61e7cda09147cdafd7839eb10be844bd35baa00c7f737","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-a1e0668357b9b581","role":"sequence","name":"basis_map_A013611 of Triangle of coefficients of Chebyshev's S(n,x+5) polynomials (exponents of x in increasing order).","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:       6,         8;\n  2:      35,        96,        48;\n  3:     204,       848,       864,       256;\n  4:    1189,      6624,     10224,      6144,       1280;\n  5:    6930,     48408,    100224,     91136,      38400,      6144;\n  6:   40391,    339264,    881568,   1075200,     684800,    221184,     28672;\n  7:  235416,   2310304,   7224768,  11061760,    9446400,   4608000,   1204224,    131072;\n  8: 1372105,  15405888,  56328864, 103827456,  111302400,  72769536,  28700672,   6291456,   589824;\n  9: 7997214, 101101864, 423183744, 912323584, 1177505280, 966260736, 510590976, 168820736, 31850496, 2621440;\n","recurrence_sentence":"The row polynomials satisfy -(8*n+3*n*x-2*n^3*x^2)*P(n,x) = -(24*n+24+(8*n^2+41*n+33)*x-(6*n^3+6*n^2-12*n-12)*x^2-(4*n^3+8*n^2+4*n)*x^3)*P(n-1,x) - (16+36*x-(8*n^2+8*n-26)*x^2-(12*n^2+12*n-6)*x^3-(4*n^2+4*n)*x^4)*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[(-(24*n + 24 + (8*n^2 + 41*n + 33)*x - (6*n^3 + 6*n^2 - 12*n - 12)*x^2 - (4*n^3 + 8*n^2 + 4*n)*x^3)*P[n - 1, x] - (16 + 36*x - (8*n^2 + 8*n - 26)*x^2 - (12*n^2 + 12*n - 6)*x^3 - (4*n^2 + 4*n)*x^4)*D[P[n - 1, x], x])/(-8*n - 3*n*x + 2*n^3*x^2)]];\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] = (-(24*n + 24 + (8*n^2 + 41*n + 33)*x - (6*n^3 + 6*n^2 - 12*n - 12)*x^2 - (4*n^3 + 8*n^2 + 4*n)*x^3)*v[i - 1] - (16 + 36*x - (8*n^2 + 8*n - 26)*x^2 - (12*n^2 + 12*n - 6)*x^3 - (4*n^2 + 4*n)*x^4)*deriv(v[i - 1], x))/(-8*n - 3*n*x + 2*n^3*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-c2b7c74f8491f01b","steps":["basis A013611","\u03b8+1"],"edges":["e-3938f5b848156b83","e-ab3c5dd4d8ad3202"],"text":"A207824 \u2192 basis A013611 \u2192 \u03b8+1"},"headline":"A207824 \u2192 basis A013611 \u2192 \u03b8+1","chain":null}