A346738 Expansion of e.g.f.: exp(exp(x) - 3*x - 1).

1, -2, 5, -13, 36, -101, 293, -848, 2523, -7365, 22402, -64395, 205285, -541802, 2057617, -3403993, 28685420, 43885023, 824532745, 4878097904, 44263112047, 357891860463, 3169228222338, 28506399763969, 266822555964441, 2573194635922990, 25606751525353741

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

Cf. A000110, A000296, A005494, A126617, A193683, A290219, A346739, A346740.

Sequence in context: A046171 A022854 A356697 * A116409 A279196 A002844

Adjacent sequences: A346735 A346736 A346737 * A346739 A346740 A346741

Keyword

sign

Author

Ilya Gutkovskiy, Jul 31 2021

Status

approved