A113237 - OEIS

%I #13 May 10 2016 17:38:37

%S 1,1,3,13,49,381,2971,26713,291873,3262969,41245651,569262981,

%T 8433896593,136060620853,2342471665899,42987065380561,838321137046081,

%U 17272648375895793,375413770580941603,8579701021461918589,205637099039964274161,5158188565847339152621

%N E.g.f.: exp(x*(1 - x^3 + x^4)/(1-x)).

%C 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.

%H Alois P. Heinz, <a href="/A113237/b113237.txt">Table of n, a(n) for n = 0..444</a>

%F 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....

%F 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

%F 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

%p a:= proc(n) option remember; `if`(n=0, 1, add(

%p       `if`(j=4, 0, a(n-j)*binomial(n-1, j-1)*j!), j=1..n))

%p     end:

%p seq(a(n), n=0..30);  # _Alois P. Heinz_, May 10 2016

%t f[n_] := n!*Sum[(-1)^k*LaguerreL[n - 4*k, -1, -1]/k!, {k, 0, Floor[n/4]}]; Table[ f[n], {n, 0, 19}]

%t 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 *)

%Y Cf. A000262, A052845, A097514, A113235, A113236.

%K nonn

%O 0,3

%A _Karol A. Penson_, Oct 19 2005