OFFSET
0,3
COMMENTS
Number of partitions of {1,..,n} into any number of lists of size not equal to 4, where a list means an ordered subset, cf. A000262.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..444
FORMULA
Expression as a sum involving generalized Laguerre polynomials, in Mathematica notation: a(n)=n!*Sum[(-1)^k*LaguerreL[n - 4*k, -1, -1]/k!, {k, 0, Floor[n/4]}], n=0, 1....
Recurrence: a(n) = (2*n-1)*a(n-1) - (n-2)*(n-1)*a(n-2) - 4*(n-3)*(n-2)*(n-1)*a(n-4) + 8*(n-4)*(n-3)*(n-2)*(n-1)*a(n-5) - 4*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*a(n-6). - Vaclav Kotesovec, Jun 24 2013
a(n) ~ n^(n-1/4)*exp(-3/2+2*sqrt(n)-n)/sqrt(2) * (1 + 187/(48*sqrt(n))). - Vaclav Kotesovec, Jun 24 2013
MAPLE
a:= proc(n) option remember; `if`(n=0, 1, add(
`if`(j=4, 0, a(n-j)*binomial(n-1, j-1)*j!), j=1..n))
end:
seq(a(n), n=0..30); # Alois P. Heinz, May 10 2016
MATHEMATICA
f[n_] := n!*Sum[(-1)^k*LaguerreL[n - 4*k, -1, -1]/k!, {k, 0, Floor[n/4]}]; Table[ f[n], {n, 0, 19}]
Range[0, 19]!* CoefficientList[ Series[ Exp[x*(1 - x^3 + x^4)/(1 - x)], {x, 0, 19}], x] (* Robert G. Wilson v, Oct 21 2005 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Karol A. Penson, Oct 19 2005
STATUS
approved