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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":"79c965f2d96f06e9107dddea32275235b6c3839a2b7f7da7b775d2cfde183d8f","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-d4d62d9c5cf9eec8","role":"sequence","name":"basis_map_A013610 of Triangle read by rows: T(n,k) = |s(n,n+1-k)|, where s(n,k) are the signed Stirling numbers of the first kind A008276 (1 <= k <= n; in other words, the unsigned Stirling numbers of the first kind in reverse 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:       2,        12;\n  2:       6,        84,        162;\n  3:      24,       552,       2349,        2592;\n  4:     120,      3912,      26649,       63072,        48600;\n  5:     720,     30672,     291924,     1089072,      1769850,       1049760;\n  6:    5040,    266544,    3285684,    16965072,     43019100,      53222832,      25719120;\n  7:   40320,   2555712,   38879676,   258386112,    900819225,    1726680240,    1727038908,     705438720;\n  8:  362880,  26872128,  487925532,  3984341184,  17796852225,   46672434576,   71948238012,   60486331392,   21427701120;\n  9: 3628800, 307912320, 6511737096, 63263837376, 346057915950, 1158665960268, 2434694350788, 3142139952384, 2281208366736, 714256704000;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = -(n^3+(12*n^3-12*n^2)*x-(81*n^3-162*n^2+81*n)*x^2)*P(n-2,x) + ((12*n^3-12*n^2)*x^2+(36*n^3-72*n^2+36*n)*x^3)*d/dx P(n-2,x) + (2*n^2-1+(12*n^2-12*n)*x)*P(n-1,x) - (12*n^2-12*n)*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 12*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 12*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + (12*n^3 - 12*n^2)*x - (81*n^3 - 162*n^2 + 81*n)*x^2)*P[n - 2, x] + ((12*n^3 - 12*n^2)*x^2 + (36*n^3 - 72*n^2 + 36*n)*x^3)*D[P[n - 2, x], x] + (2*n^2 - 1 + (12*n^2 - 12*n)*x)*P[n - 1, x] - (12*n^2 - 12*n)*x^2*D[P[n - 1, x], x])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 + (12*n^3 - 12*n^2)*x - (81*n^3 - 162*n^2 + 81*n)*x^2)*v[i - 2] + ((12*n^3 - 12*n^2)*x^2 + (36*n^3 - 72*n^2 + 36*n)*x^3)*deriv(v[i - 2], x) + (2*n^2 - 1 + (12*n^2 - 12*n)*x)*v[i - 1] - (12*n^2 - 12*n)*x^2*deriv(v[i - 1], x))/(n - 1)); 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-4d60c3dd8627c01d","steps":["basis A013610","(\u03b8+1)\u00b2"],"edges":["e-af0dc46429e1a2ef","e-ae82fb5b486c3eec"],"text":"A094638 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2"},"headline":"A094638 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2","chain":null}