A343884 Expansion of e.g.f. exp( x/(1-x)^2 ) / (1-x)^2.

1, 3, 15, 103, 885, 9051, 106843, 1425495, 21166953, 345678355, 6150501831, 118313311623, 2444917863325, 53982840948843, 1267645359117075, 31531781398100791, 827910838693667793, 22874802838645217955, 663243613324249850623, 20130710499843811837095

Offset: 0

Author

Seiichi Manyama, Mar 18 2023

Keywords

• nonn

Links

• Winston de Greef, Table of n, a(n) for n = 0..430

Crossrefs

Column k=2 of A361616.

Cf. A000262, A361598.

Programs

Mathematica

Table[n!*Sum[Binomial[n + k + 1, n - k]/k!, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 25 2023 *)

PARI

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x/(1-x)^2)/(1-x)^2))

PARI

a(n) = n! * sum(k=0, n, binomial(n+k+1,n-k)/k!) \\ Winston de Greef, Mar 19 2023

Formula

a(n) = n! * Sum_{k=0..n} binomial(n+k+1,n-k)/k! = Sum_{k=0..n} (n+k+1)!/(2*k+1)! * binomial(n,k).

From Vaclav Kotesovec, Mar 25 2023: (Start)

a(n) ~ exp(-1/12 + 3*2^(-2/3)*n^(2/3) - n) * n^(n + 1/2) / sqrt(6) * (1 + 2^(1/3)/n^(1/3) + 323/(360*2^(1/3)*n^(2/3))).

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