A295610 a(n) = Sum_{k=0..n} (n!/(n - k)!)^k.

1, 2, 7, 256, 345749, 25090776406, 139507578065088907, 82622801516492599819822772, 6985137485409222182920705065038896201, 109110989095384931538566720095053550173384985449034, 395940975233113726268241745444050219538058574725338743701918216111

Offset: 0

Author

Ilya Gutkovskiy, Nov 24 2017

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..30
• Eric Weisstein's World of Mathematics, Binomial Sums

Crossrefs

Cf. A000522, A006040, A036740, A217284, A219206.

Programs

Mathematica

Table[Sum[(n!/(n - k)!)^k, {k, 0, n}], {n, 0, 10}]
Table[Sum[(Gamma[n + 1]/Gamma[k + 1])^(n - k), {k, 0, n}], {n, 0, 10}]
Table[Sum[(Binomial[n, k] k!)^k, {k, 0, n}], {n, 0, 10}]

PARI

a(n) = sum(k=0, n, (n!/(n - k)!)^k); \\ Michel Marcus, Nov 25 2017

Formula

a(n) = Sum_{k=0..n} A219206(n,k)*A036740(k).

a(n) ~ 2^(n/2) * Pi^(n/2) * n^(n^2 + n/2) / exp(n^2 - 1/12). - Vaclav Kotesovec, Nov 25 2017