A361598 Expansion of e.g.f. exp( x/(1-x)^2 ) / (1-x).

1, 2, 9, 58, 473, 4626, 52537, 677594, 9762993, 155175778, 2693718281, 50657791482, 1025158123849, 22198908725618, 511885585833273, 12517101011344666, 323402336324055137, 8800318580852865474, 251497162228635927433, 7529081846683064675258

Offset: 0

Author

Seiichi Manyama, Mar 17 2023

Keywords

• nonn

Crossrefs

Column k=2 of A361600.

Cf. A082579.

Programs

Mathematica

Table[n! * Sum[Binomial[n+k,2*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)^2)/(1-x)))

Formula

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

From Vaclav Kotesovec, Mar 17 2023: (Start)

a(n) = (3*n - 1)*a(n-1) - (n-1)*(3*n - 5)*a(n-2) + (n-2)^2*(n-1)*a(n-3).

a(n) ~ 2^(-1/6) * 3^(-1/2) * exp(-1/12 + 3*2^(-2/3)*n^(2/3) - n) * n^(n + 1/6) * (1 + 1/(2^(2/3)*n^(1/3)) + 83/(360*2^(1/3)*n^(2/3))). (End)