A346739 Expansion of e.g.f.: exp(exp(x) - 4*x - 1).

1, -3, 10, -35, 127, -472, 1787, -6855, 26572, -103765, 407695, -1608378, 6369117, -25271183, 100542930, -400114103, 1597052419, -6359524256, 25481982047, -101103395443, 409291679676, -1592903606657, 6729506287091, -23748796926026, 123501587468073, -227183793907851

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..580

Formula

G.f. A(x) satisfies: A(x) = (1 - x + x * A(x/(1 - x))) / ((1 - x) * (1 + 4*x)).

a(n) = Sum_{k=0..n} binomial(n,k) * (-4)^(n-k) * Bell(k).

a(n) = exp(-1) * Sum_{k>=0} (k - 4)^n / k!.

a(0) = 1; a(n) = -4 * a(n-1) + Sum_{k=0..n-1} binomial(n-1,k) * a(k).

Mathematica

nmax = 25; CoefficientList[Series[Exp[Exp[x] - 4 x - 1], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Binomial[n, k] (-4)^(n - k) BellB[k], {k, 0, n}], {n, 0, 25}]
a[0] = 1; a[n_] := a[n] = -4 a[n - 1] + Sum[Binomial[n - 1, k] a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 25}]

Prog

(Magma)
R<x>:=PowerSeriesRing(Rationals(), 30);
Coefficients(R!(Laplace( Exp(Exp(x) -4*x -1) ))) // G. C. Greubel, Jun 12 2024
(SageMath)
[factorial(n)*( exp(exp(x) -4*x -1) ).series(x, n+1).list()[n] for n in (0..30)] # G. C. Greubel, Jun 12 2024

Crossrefs

Cf. A000110, A000296, A045379, A126617, A193684, A290219, A346738, A346740.

Sequence in context: A316596 A047127 A114196 * A078789 A299443 A128736

Adjacent sequences: A346736 A346737 A346738 * A346740 A346741 A346742

Keyword

sign

Author

Ilya Gutkovskiy, Jul 31 2021

Status

approved