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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":"58fa0c571f7f07fd504b559e13a489cad299ac62f97db44ac1f5802b5dafaa45","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-107a00b591d3c4c8","role":"sequence","name":"basis_map_A013609 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,       36;\n  3:    48,     352,      432,      128;\n  4:   164,    1664,     3240,     2048,      400;\n  5:   560,    7264,    19584,    19456,     8000,     1152;\n  6:  1912,   30208,   104544,   143360,    92000,    27648,     3136;\n  7:  6528,  121600,   514944,   906240,   800000,   373248,    87808,    8192;\n  8: 22288,  478208,  2397888,  5169152,  5848000,  3741696,  1361024,  262144,  20736;\n  9: 76096, 1847936, 10713600, 27389952, 37945600, 31062528, 15454208, 4587520, 746496, 51200;\n","recurrence_sentence":"The row polynomials satisfy (16*n+4*n*x-(20*n^3-4*n)*x^2-10*n^3*x^3+n^5*x^4)*P(n,x) = (32*n+32+(28*n^2+72*n+44)*x-(40*n^3+44*n^2-24*n-28)*x^2-(4*n^4+54*n^3+84*n^2+30*n-4)*x^3+(2*n^5+2*n^4-18*n^3-36*n^2-22*n-4)*x^4+(n^5+3*n^4+3*n^3+n^2)*x^5)*P(n-1,x) + (8+24*x-(28*n^2+28*n-20)*x^2-(76*n^2+76*n)*x^3+(2*n^4+4*n^3-52*n^2-54*n-8)*x^4+(4*n^4+8*n^3-6*n^2-10*n-2)*x^5+(n^4+2*n^3+n^2)*x^6)*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 + 32 + (28*n^2 + 72*n + 44)*x - (40*n^3 + 44*n^2 - 24*n - 28)*x^2 - (4*n^4 + 54*n^3 + 84*n^2 + 30*n - 4)*x^3 + (2*n^5 + 2*n^4 - 18*n^3 - 36*n^2 - 22*n - 4)*x^4 + (n^5 + 3*n^4 + 3*n^3 + n^2)*x^5)*P[n - 1, x] + (8 + 24*x - (28*n^2 + 28*n - 20)*x^2 - (76*n^2 + 76*n)*x^3 + (2*n^4 + 4*n^3 - 52*n^2 - 54*n - 8)*x^4 + (4*n^4 + 8*n^3 - 6*n^2 - 10*n - 2)*x^5 + (n^4 + 2*n^3 + n^2)*x^6)*D[P[n - 1, x], x])/(16*n + 4*n*x - (20*n^3 - 4*n)*x^2 - 10*n^3*x^3 + n^5*x^4)]];\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 + 32 + (28*n^2 + 72*n + 44)*x - (40*n^3 + 44*n^2 - 24*n - 28)*x^2 - (4*n^4 + 54*n^3 + 84*n^2 + 30*n - 4)*x^3 + (2*n^5 + 2*n^4 - 18*n^3 - 36*n^2 - 22*n - 4)*x^4 + (n^5 + 3*n^4 + 3*n^3 + n^2)*x^5)*v[i - 1] + (8 + 24*x - (28*n^2 + 28*n - 20)*x^2 - (76*n^2 + 76*n)*x^3 + (2*n^4 + 4*n^3 - 52*n^2 - 54*n - 8)*x^4 + (4*n^4 + 8*n^3 - 6*n^2 - 10*n - 2)*x^5 + (n^4 + 2*n^3 + n^2)*x^6)*deriv(v[i - 1], x))/(16*n + 4*n*x - (20*n^3 - 4*n)*x^2 - 10*n^3*x^3 + n^5*x^4)); 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 A013609","(\u03b8+1)\u00b2"],"edges":["e-3133312db52cd60e","e-bea4235969fc8bee"],"text":"A110441 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2"},"headline":"A110441 \u2192 basis A013609 \u2192 (\u03b8+1)\u00b2","chain":null}