cp's OEIS Frontend

A088991 Derangement numbers d(n,4) 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.

%I A088991 #17 Apr 24 2025 09:44:48
%S A088991 1,0,4,32,432,7424,157120,3949056,114972928,3805503488,141137150976,
%T A088991 5797706178560,261309106499584,12821127008550912,680286677982625792,
%U A088991 38814037079505895424,2369659425449311272960,154142301601844298776576,10642813349855965483368448
%N A088991 Derangement numbers d(n,4) 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.
%H A088991 Vincenzo Librandi, <a href="/A088991/b088991.txt">Table of n, a(n) for n = 0..200</a>
%F A088991 Inverse binomial transform of A007696. E.g.f.: exp(-x)/(1-4*x)^(1/4). - _Vladeta Jovovic_, Dec 17 2003
%F A088991 a(n) ~ n^(n-1/4) * Gamma(3/4) * 4^n / (sqrt(Pi)*exp(n+1/4)). - _Vaclav Kotesovec_, Oct 08 2013
%F A088991 From _Seiichi Manyama_, Apr 23 2025: (Start)
%F A088991 E.g.f.: B(x)^4, where B(x) is the e.g.f. of A381504.
%F A088991 a(n) = (-1)^n * n! * Sum_{k=0..n} 4^k * binomial(-1/4,k)/(n-k)!. (End)
%t A088991 CoefficientList[Series[E^(-x)/(1-4*x)^(1/4), {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Oct 08 2013 *)
%Y A088991 Cf. A000166, A053871, A033030, A088992.
%Y A088991 Cf. A381504.
%K A088991 nonn,easy
%O A088991 0,3
%A A088991 _N. J. A. Sloane_, Nov 02 2003