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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":"e0f032a34ff72d9bfe74d454ed62e5816d5f677f30204960be4f08d908701f85","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-35273d9764d27bc4","role":"sequence","name":"basis_map_A013611 of Triangular array formed by the Mersenne numbers.","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,       8;\n  2:    14,      64,       48;\n  3:    48,     352,      576,      256;\n  4:   164,    1664,     4320,     4096,      1280;\n  5:   560,    7264,    26112,    38912,     25600,      6144;\n  6:  1912,   30208,   139392,   286720,    294400,    147456,     28672;\n  7:  6528,  121600,   686592,  1812480,   2560000,   1990656,    802816,   131072;\n  8: 22288,  478208,  3197184, 10338304,  18713600,  19955712,  12443648,  4194304,   589824;\n  9: 76096, 1847936, 14284800, 54779904, 121425920, 165666816, 141295616, 73400320, 21233664, 2621440;\n","recurrence_sentence":"The row polynomials satisfy -(2*n+2*n*x-2*n^3*x^2)*P(n,x) = -(4*n+4+(2*n^2+12*n+10)*x-(4*n^3+4*n^2-8*n-8)*x^2-(4*n^3+8*n^2+4*n)*x^3)*P(n-1,x) - (1+6*x-(2*n^2+2*n-10)*x^2-(8*n^2+8*n-4)*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[(-(4*n + 4 + (2*n^2 + 12*n + 10)*x - (4*n^3 + 4*n^2 - 8*n - 8)*x^2 - (4*n^3 + 8*n^2 + 4*n)*x^3)*P[n - 1, x] - (1 + 6*x - (2*n^2 + 2*n - 10)*x^2 - (8*n^2 + 8*n - 4)*x^3 - (4*n^2 + 4*n)*x^4)*D[P[n - 1, x], x])/(-2*n - 2*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] = (-(4*n + 4 + (2*n^2 + 12*n + 10)*x - (4*n^3 + 4*n^2 - 8*n - 8)*x^2 - (4*n^3 + 8*n^2 + 4*n)*x^3)*v[i - 1] - (1 + 6*x - (2*n^2 + 2*n - 10)*x^2 - (8*n^2 + 8*n - 4)*x^3 - (4*n^2 + 4*n)*x^4)*deriv(v[i - 1], x))/(-2*n - 2*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-9b5c7e97e5191da6","steps":["basis A013611","\u03b8+1"],"edges":["e-6b7a7c8afbb896b5","e-ab8af86e6cae2998"],"text":"A110441 \u2192 basis A013611 \u2192 \u03b8+1"},"headline":"A110441 \u2192 basis A013611 \u2192 \u03b8+1","chain":null}