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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":"e38f6316f355bb9bc6ceb484cfabc435900a7bf04da3b8def5ad3eba96ae291e","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-125f84bee6f03b65","role":"sequence","name":"one_plus_derivative of Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.","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:     1,      2;\n  2:     1,      6,       3;\n  3:     2,     16,      18,       4;\n  4:     6,     56,      87,      40,      5;\n  5:    24,    248,     465,     300,     75,      6;\n  6:   120,   1336,    2847,    2260,    800,    126,     7;\n  7:   720,   8496,   19908,   18256,   8050,   1806,   196,    8;\n  8:  5040,  62352,  157392,  160832,  82845,  23352,  3626,  288,   9;\n  9: 40320, 518976, 1391868, 1549040, 897645, 297990, 58506, 6672, 405, 10;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n^3-9*n^2+27*n-27+(3*n^2-18*n+27)*x+(3*n-9)*x^2+x^3)*P(n-3,x) - (3*n^2-15*n+19+(6*n-15)*x+3*x^2)*P(n-2,x) + (3*n-6+3*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 1 + 2*x, P(2,x) = 1 + 6*x + 3*x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 2*x;\nP[2, x_] := 1 + 6*x + 3*x^2;\nP[n_, x_] := P[n, x] = Expand[(n^3 - 9*n^2 + 27*n - 27 + (3*n^2 - 18*n + 27)*x + (3*n - 9)*x^2 + x^3)*P[n - 3, x] - (3*n^2 - 15*n + 19 + (6*n - 15)*x + 3*x^2)*P[n - 2, x] + (3*n - 6 + 3*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 2*x; v[3] = 1 + 6*x + 3*x^2; for(n = 3, N, my(i = n + 1); v[i] = (n^3 - 9*n^2 + 27*n - 27 + (3*n^2 - 18*n + 27)*x + (3*n - 9)*x^2 + x^3)*v[i - 3] - (3*n^2 - 15*n + 19 + (6*n - 15)*x + 3*x^2)*v[i - 2] + (3*n - 6 + 3*x)*v[i - 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-b9fbfb57d9269e2f","steps":["D+1","\u03b8+1"],"edges":["e-649ce40fa3feb4c2","e-03b2d53ee3f71db6"],"text":"A132393 \u2192 D+1 \u2192 \u03b8+1"},"headline":"A132393 \u2192 D+1 \u2192 \u03b8+1","chain":null}