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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":"de31cc67233d7e7430fefbbb627164d92e8fd97baacf751a6dd317f34c5a596d","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-0164479e47952770","role":"sequence","name":"Triangle read by rows. Stirling set numbers of order 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k > n, otherwise T(n, k) = T(n-1, k-1) + k^2*T(n-1, k), for 0 <= k <= n.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 0, 2;\n   3: 0, 2,     3;\n   4: 0, 2,    15,       4;\n   5: 0, 2,    63,      56,        5;\n   6: 0, 2,   255,     588,      150,        6;\n   7: 0, 2,  1023,    5632,     3135,      330,       7;\n   8: 0, 2,  4095,   52052,    57200,    12012,     637,     8;\n   9: 0, 2, 16383,  473928,   980265,   368940,   36946,  1120,    9;\n  10: 0, 2, 65535, 4287196, 16276650, 10399818, 1760486, 97104, 1836, 10;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(2*x+x^2)*P(n-2,x) - x*d/dx P(n-2,x) - (x^2+x^3)*d^2/dx^2 P(n-2,x) + (1+2*x)*P(n-1,x) + x*d/dx P(n-1,x) + x^2*d^2/dx^2 P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 2*x;\nP[n_, x_] := P[n, x] = Expand[-(2*x + x^2)*P[n - 2, x] - x*D[P[n - 2, x], x] - (x^2 + x^3)*D[P[n - 2, x], {x, 2}] + (1 + 2*x)*P[n - 1, x] + x*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 2}]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2*x; for(n = 3, N, my(i = n); v[i] = -(2*x + x^2)*v[i - 2] - x*deriv(v[i - 2], x) - (x^2 + x^3)*deriv(deriv(v[i - 2], x), x) + (1 + 2*x)*v[i - 1] + x*deriv(v[i - 1], x) + x^2*deriv(deriv(v[i - 1], x), x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-0164479e47952770","steps":["\u03b8+1"],"edges":["e-37fc17f560d06793"],"text":"A269945 \u2192 \u03b8+1"},"headline":"A269945 \u2192 \u03b8+1","chain":null}