A088992 Derangement numbers d(n,5) where d(n,k) = k(n-1)(d(n-1,k) + d(n-2,k)), with d(0,k) = 1 and d(1,k) = 0.

1, 0, 5, 50, 825, 17500, 458125, 14268750, 515440625, 21188375000, 976671703125, 49893003906250, 2797832158515625, 170863509745312500, 11287987223748828125, 802119551344589843750, 61005565392625400390625, 4944614795517599218750000

Offset: 0

Author

N. J. A. Sloane, Nov 02 2003

Keywords

• nonn
• easy

Comments

In general, d(n,k) is asymptotic to sqrt(2*Pi) * k^n * n^(n + 1/2) / (Gamma(1/k) * exp((n*k+1)/k) * n^((k-1)/k)), for k>0. - Vaclav Kotesovec, Oct 31 2017

Crossrefs

Cf. A000166, A053871, A033030, A088991.

Formula

Inverse binomial transform of A008548. E.g.f.: exp(-x)/(1-5*x)^(1/5). - Vladeta Jovovic, Dec 17 2003

a(n) ~ Pi * sqrt(2) * n^(n-3/10) * 5^n / (sqrt(Pi) * Gamma(1/5) * exp(n + 1/5)). - Vaclav Kotesovec, Oct 31 2017

a(n) = (-1)^n * n! * Sum_{k=0..n} 5^k * binomial(-1/5,k)/(n-k)!. - Seiichi Manyama, Apr 23 2025