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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":"2963a3742003fa5b114905fa59d00b464edd2f85301cb6c752952540f96b1233","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-c77d5d4d44ca1f43","role":"sequence","name":"Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 3*x + 2.","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,       4;\n  2:   4,      40,        12;\n  3:   8,     264,       396,         32;\n  4:  16,    1496,      6336,       2992,          80;\n  5:  32,    7928,     75372,     100496,       19820,         192;\n  6:  64,   40728,    769464,    2210912,     1282440,      122184,         448;\n  7: 128,  206200,   7194276,   38683360,    48354200,    14388552,      721700,       1024;\n  8: 256, 1036888,  63740208,  588587216,  1353917600,   882880824,   148727152,    4147552,     2304;\n  9: 512, 5197752, 545694300, 8174198256, 31462324740, 37754789688, 14304846948, 1455184800, 23389884, 5120;\n","recurrence_sentence":"The row polynomials satisfy (3*n-1)*P(n,x) = (6*n+4+(18*n^2-12*n-16)*x-(27*n^3-27*n^2-18*n+8)*x^2)*P(n-2,x) + ((9*n+6)*x-(27*n^2+18*n)*x^2+(27*n^2+9*n-6)*x^3)*d/dx P(n-2,x) + (3*n-4+(18*n^2-12*n+2)*x)*P(n-1,x) + ((9*n-3)*x-(9*n-3)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[((6*n + 4 + (18*n^2 - 12*n - 16)*x - (27*n^3 - 27*n^2 - 18*n + 8)*x^2)*P[n - 2, x] + ((9*n + 6)*x - (27*n^2 + 18*n)*x^2 + (27*n^2 + 9*n - 6)*x^3)*D[P[n - 2, x], x] + (3*n - 4 + (18*n^2 - 12*n + 2)*x)*P[n - 1, x] + ((9*n - 3)*x - (9*n - 3)*x^2)*D[P[n - 1, x], x])/(3*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 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((6*n + 4 + (18*n^2 - 12*n - 16)*x - (27*n^3 - 27*n^2 - 18*n + 8)*x^2)*v[i - 2] + ((9*n + 6)*x - (27*n^2 + 18*n)*x^2 + (27*n^2 + 9*n - 6)*x^3)*deriv(v[i - 2], x) + (3*n - 4 + (18*n^2 - 12*n + 2)*x)*v[i - 1] + ((9*n - 3)*x - (9*n - 3)*x^2)*deriv(v[i - 1], x))/(3*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-c77d5d4d44ca1f43","steps":["\u03b8+1"],"edges":["e-5ce18864d471ed1f"],"text":"A257610 \u2192 \u03b8+1"},"headline":"A257610 \u2192 \u03b8+1","chain":null}