A367891 Expansion of e.g.f. exp(4*(exp(x) - 1 - x)).

1, 0, 4, 4, 52, 164, 1364, 7620, 60148, 449252, 3831700, 33811716, 320082228, 3178774564, 33234163668, 363535920196, 4153091085172, 49406896240996, 610777358429204, 7830140410294148, 103914148870277556, 1425254885630973604, 20173671034640405588

Offset

0,3

Links

Table of n, a(n) for n=0..22.

Formula

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

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

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

a(n) = Sum_{k=0..n} binomial(n,k) * (-4)^(n-k) * A078944(k).

Mathematica

nmax = 22; CoefficientList[Series[Exp[4 (Exp[x] - 1 - x)], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = 4 Sum[Binomial[n - 1, k] a[n - k - 1], {k, 1, n - 1}]; Table[a[n], {n, 0, 22}]
Table[Sum[Binomial[n, k] (-4)^(n - k) BellB[k, 4], {k, 0, n}], {n, 0, 22}]

Prog

(PARI) my(x='x+O('x^30)); Vec(serlaplace(exp(4*(exp(x) - 1 - x)))) \\ Michel Marcus, Dec 04 2023

Crossrefs

Cf. A000296, A078944, A194689, A346739, A367890.

Sequence in context: A189102 A220778 A364763 * A322099 A009644 A196180

Adjacent sequences: A367888 A367889 A367890 * A367892 A367893 A367894

Keyword

nonn

Author

Ilya Gutkovskiy, Dec 04 2023

Status

approved