OFFSET
0,2
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..579
FORMULA
G.f. A(x) satisfies: A(x) = (1 - x + x * A(x/(1 - x))) / ((1 - x) * (1 + 3*x)).
a(n) = Sum_{k=0..n} binomial(n,k) * (-3)^(n-k) * Bell(k).
a(n) = exp(-1) * Sum_{k>=0} (k - 3)^n / k!.
a(0) = 1; a(n) = -3 * a(n-1) + Sum_{k=0..n-1} binomial(n-1,k) * a(k).
MATHEMATICA
nmax = 26; CoefficientList[Series[Exp[Exp[x] - 3 x - 1], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Binomial[n, k] (-3)^(n - k) BellB[k], {k, 0, n}], {n, 0, 26}]
a[0] = 1; a[n_] := a[n] = -3 a[n - 1] + Sum[Binomial[n - 1, k] a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 26}]
PROG
(Magma)
R<x>:=PowerSeriesRing(Rationals(), 50);
Coefficients(R!(Laplace( Exp(Exp(x)-3*x-1) ))) // G. C. Greubel, Jun 12 2024
(SageMath)
[factorial(n)*( exp(exp(x)-3*x-1) ).series(x, n+1).list()[n] for n in (0..30)] # G. C. Greubel, Jun 12 2024
CROSSREFS
KEYWORD
sign
AUTHOR
Ilya Gutkovskiy, Jul 31 2021
STATUS
approved