A070227 a(n) = Sum_{k>=0} (k+n)!^n/((k+n)!*(k!^n)*exp(1)).

1, 1, 7, 2971, 326922081, 25571928251231076, 3104750712141723393459934903, 1106411839720559249283387766293758050197271, 1982711933502621451047063899803427489760228712955842831202561

Offset: 0

Author

Benoit Cloitre, May 07 2002

Keywords

• easy
• nonn

Links

• Amiram Eldar, Table of n, a(n) for n = 0..24

Crossrefs

Cf. A090210.

Programs

Mathematica

a[n_] := Exp[-1] * Sum[(k + n)!^n/((k + n)!*(k!^n)), {k, 0, Infinity}]; Array[a, 10, 0] (* Amiram Eldar, May 01 2025 *)
Table[Sum[(-1)^(k+m) * m!^(n-1) / ((k-m)!*(m-n)!^n), {k, n, n^2}, {m, n, k}], {n, 0, 10}] (* Vaclav Kotesovec, May 01 2025 *)

PARI

\\ precision 1000 digits :
for(n=1,9,print1(round(sum(k=0,200,(k+n)!^n/((k+n)!*(k!^n)*exp(1)))),","))

Formula

a(n) = A090210(n,n). - Alois P. Heinz, Aug 01 2016

a(n) = Sum_{k=n..n^2} Sum_{m=n..k} (-1)^(k+m) * m!^(n-1) / ((k-m)! * (m-n)!^n). - Vaclav Kotesovec, May 01 2025

Extensions

a(0)=1 prepended by Alois P. Heinz, Aug 01 2016