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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":"21aa2128969a3444273f8edea06a0053cac5ad62514b3e40e5d726f654c730fe","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-dd70177049267f47","role":"sequence","name":"basis_map_A007318 of Triangle of coefficients of polynomials u(n,x) jointly generated with A207607; see the Formula section.","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:    13,     22,       6;\n  1:    34,     88,      51,       8;\n  2:    89,    310,     279,      92,      10;\n  3:   233,   1020,    1251,     640,     145,      12;\n  4:   610,   3216,    5004,    3496,    1225,     210,      14;\n  5:  1597,   9848,   18585,   16520,    7900,    2088,     287,     16;\n  6:  4181,  29522,   65523,   70844,   43125,   15534,    3283,    376,    18;\n  7: 10946,  87080,  222267,  283376,  210030,   96264,   27685,   4864,   477,   20;\n  8: 28657, 253610,  731898, 1075640,  941185,  525294,  192080,  45856,  6885,  590,  22;\n  9: 75025, 731064, 2353842, 3919408, 3958075, 2609040, 1161398, 352224, 71766, 9400, 715, 24;\n","recurrence_sentence":"The row polynomials satisfy -(20*n+(66*n+90)*x-(2*n^3-48*n-66)*x^2-(4*n^3+18*n^2+26*n+12)*x^3)*P(n,x) = -(30*n-20+(5*n^2+119*n+134)*x-(3*n^3-12*n^2-183*n-266)*x^2-(7*n^3+29*n^2-2*n-64)*x^3-(2*n^3+12*n^2+22*n+12)*x^4)*P(n-1,x) - (25+145*x-(5*n^2+5*n-243)*x^2-(16*n^2+46*n-113)*x^3-(13*n^2+49*n+16)*x^4-(2*n^2+8*n+6)*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 13 + 22*x + 6*x^2.","mathematica":"P[0, x_] := 13 + 22*x + 6*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(30*n - 20 + (5*n^2 + 119*n + 134)*x - (3*n^3 - 12*n^2 - 183*n - 266)*x^2 - (7*n^3 + 29*n^2 - 2*n - 64)*x^3 - (2*n^3 + 12*n^2 + 22*n + 12)*x^4)*P[n - 1, x] - (25 + 145*x - (5*n^2 + 5*n - 243)*x^2 - (16*n^2 + 46*n - 113)*x^3 - (13*n^2 + 49*n + 16)*x^4 - (2*n^2 + 8*n + 6)*x^5)*D[P[n - 1, x], x])/(-20*n - (66*n + 90)*x + (2*n^3 - 48*n - 66)*x^2 + (4*n^3 + 18*n^2 + 26*n + 12)*x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 13 + 22*x + 6*x^2; for(n = 1, N, my(i = n + 1); v[i] = (-(30*n - 20 + (5*n^2 + 119*n + 134)*x - (3*n^3 - 12*n^2 - 183*n - 266)*x^2 - (7*n^3 + 29*n^2 - 2*n - 64)*x^3 - (2*n^3 + 12*n^2 + 22*n + 12)*x^4)*v[i - 1] - (25 + 145*x - (5*n^2 + 5*n - 243)*x^2 - (16*n^2 + 46*n - 113)*x^3 - (13*n^2 + 49*n + 16)*x^4 - (2*n^2 + 8*n + 6)*x^5)*deriv(v[i - 1], x))/(-20*n - (66*n + 90)*x + (2*n^3 - 48*n - 66)*x^2 + (4*n^3 + 18*n^2 + 26*n + 12)*x^3)); 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-dfab35176bd08e72","steps":["x \u21a6 x+1","\u03b8+1"],"edges":["e-23e971f1ed1ce0e7","e-494a1ff081b91da5"],"text":"A207606 \u2192 x \u21a6 x+1 \u2192 \u03b8+1"},"headline":"A207606 \u2192 x \u21a6 x+1 \u2192 \u03b8+1","chain":null}