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.
1, 0, 4, 32, 432, 7424, 157120, 3949056, 114972928, 3805503488, 141137150976, 5797706178560, 261309106499584, 12821127008550912, 680286677982625792, 38814037079505895424, 2369659425449311272960, 154142301601844298776576, 10642813349855965483368448
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..200
Programs
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Mathematica
CoefficientList[Series[E^(-x)/(1-4*x)^(1/4), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 08 2013 *)
Formula
Inverse binomial transform of A007696. E.g.f.: exp(-x)/(1-4*x)^(1/4). - Vladeta Jovovic, Dec 17 2003
a(n) ~ n^(n-1/4) * Gamma(3/4) * 4^n / (sqrt(Pi)*exp(n+1/4)). - Vaclav Kotesovec, Oct 08 2013
From Seiichi Manyama, Apr 23 2025: (Start)
E.g.f.: B(x)^4, where B(x) is the e.g.f. of A381504.
a(n) = (-1)^n * n! * Sum_{k=0..n} 4^k * binomial(-1/4,k)/(n-k)!. (End)