A337040 a(n) = exp(-1/4) * Sum_{k>=0} (4*k - 1)^n / (4^k * k!).

1, 0, 4, 16, 112, 896, 8384, 88320, 1032448, 13242368, 184591360, 2773929984, 44641579008, 765196926976, 13905753980928, 266855007453184, 5388980396818432, 114172599765827584, 2530858142594760704, 58556990344729198592, 1411095950792925904896, 35347148031264582270976

Offset

0,3

Links

Table of n, a(n) for n=0..21.

Formula

G.f. A(x) satisfies: A(x) = (1 - 4*x + x*A(x/(1 - 4*x))) / (1 - 3*x - 4*x^2).

G.f.: (1/(1 + x)) * Sum_{k>=0} (x/(1 + x))^k / Product_{j=1..k} (1 - 4*j*x/(1 + x)).

E.g.f.: exp((exp(4*x) - 1) / 4 - x).

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

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * A004213(k).

a(n) ~ 4^(n - 1/4) * n^(n - 1/4) * exp(n/LambertW(4*n) - n - 1/4) / (sqrt(1 + LambertW(4*n)) * LambertW(4*n)^(n - 1/4)). - Vaclav Kotesovec, Jun 26 2022

Mathematica

nmax = 21; CoefficientList[Series[Exp[(Exp[4 x] - 1)/4 - x], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k] 4^k a[n - k - 1], {k, 1, n - 1}]; Table[a[n], {n, 0, 21}]
Table[Sum[(-1)^(n - k) Binomial[n, k] 4^k BellB[k, 1/4], {k, 0, n}], {n, 0, 21}]

Crossrefs

Cf. A000296, A003576, A004213, A337038, A337039, A337041, A337042, A337043.

Sequence in context: A332752 A356130 A210573 * A339200 A087335 A204573

Adjacent sequences: A337037 A337038 A337039 * A337041 A337042 A337043

Keyword

nonn

Author

Ilya Gutkovskiy, Aug 12 2020

Status

approved