A361599 Expansion of e.g.f. exp( x/(1-x)^3 ) / (1-x).

1, 2, 11, 88, 881, 10526, 145867, 2294636, 40302593, 780263866, 16483592171, 376901809472, 9265228770481, 243493769839958, 6808261249400171, 201697053847178836, 6308214318127014017, 207622266953125336946, 7170928402389293540683, 259247888385780787392296

Offset: 0

Author

Seiichi Manyama, Mar 17 2023

Keywords

• nonn

Crossrefs

Column k=3 of A361600.

Cf. A091695.

Programs

Mathematica

Table[n! * Sum[Binomial[n+2*k,3*k]/k!, {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Mar 17 2023 *)

PARI

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x/(1-x)^3)/(1-x)))

Formula

a(n) = n! * Sum_{k=0..n} binomial(n+2*k,3*k)/k! = Sum_{k=0..n} (n+2*k)!/(3*k)! * binomial(n,k).

From Vaclav Kotesovec, Mar 17 2023: (Start)

a(n) = 2*(2*n - 1)*a(n-1) - (n-1)*(6*n - 11)*a(n-2) + (n-2)*(n-1)*(4*n - 9)*a(n-3) - (n-3)^2*(n-2)*(n-1)*a(n-4).

a(n) ~ 3^(-1/8) * exp(-1/27 - 3^(-5/4)*n^(1/4)/8 + sqrt(n/3)/2 + 4*3^(-3/4)*n^(3/4) - n) * n^(n + 1/8) / 2 * (1 + (34237/69120)*3^(1/4)/n^(1/4)). (End)