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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":"f38105bab30aff795c0d8e7305e5fffc2b7696cc71b613fc0941444a33538418","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-89d60963344cadbc","role":"sequence","name":"basis_map_A013610 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,       6;\n  2:    14,      48,      27;\n  3:    48,     264,     324,      108;\n  4:   164,    1248,    2430,     1728,      405;\n  5:   560,    5448,   14688,    16416,     8100,     1458;\n  6:  1912,   22656,   78408,   120960,    93150,    34992,     5103;\n  7:  6528,   91200,  386208,   764640,   810000,   472392,   142884,   17496;\n  8: 22288,  358656, 1798416,  4361472,  5921100,  4735584,  2214702,  559872,   59049;\n  9: 76096, 1385952, 8035200, 23110272, 38419920, 39313512, 25147584, 9797760, 2125764, 196830;\n","recurrence_sentence":"The row polynomials satisfy -(96*n+72*n*x-54*n^3*x^2)*P(n,x) = -(192*n+192+(72*n^2+432*n+360)*x-(108*n^3+108*n^2-216*n-216)*x^2-(81*n^3+162*n^2+81*n)*x^3)*P(n-1,x) - (64+288*x-(72*n^2+72*n-360)*x^2-(216*n^2+216*n-108)*x^3-(81*n^2+81*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[(-(192*n + 192 + (72*n^2 + 432*n + 360)*x - (108*n^3 + 108*n^2 - 216*n - 216)*x^2 - (81*n^3 + 162*n^2 + 81*n)*x^3)*P[n - 1, x] - (64 + 288*x - (72*n^2 + 72*n - 360)*x^2 - (216*n^2 + 216*n - 108)*x^3 - (81*n^2 + 81*n)*x^4)*D[P[n - 1, x], x])/(-96*n - 72*n*x + 54*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] = (-(192*n + 192 + (72*n^2 + 432*n + 360)*x - (108*n^3 + 108*n^2 - 216*n - 216)*x^2 - (81*n^3 + 162*n^2 + 81*n)*x^3)*v[i - 1] - (64 + 288*x - (72*n^2 + 72*n - 360)*x^2 - (216*n^2 + 216*n - 108)*x^3 - (81*n^2 + 81*n)*x^4)*deriv(v[i - 1], x))/(-96*n - 72*n*x + 54*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 A013610","\u03b8+1"],"edges":["e-5de408358bde06b7","e-825b334db1f8b2f3"],"text":"A110441 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A110441 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}