A113237 E.g.f.: exp(x*(1 - x^3 + x^4)/(1-x)).

1, 1, 3, 13, 49, 381, 2971, 26713, 291873, 3262969, 41245651, 569262981, 8433896593, 136060620853, 2342471665899, 42987065380561, 838321137046081, 17272648375895793, 375413770580941603, 8579701021461918589, 205637099039964274161, 5158188565847339152621

Offset: 0

Author

Karol A. Penson, Oct 19 2005

Keywords

• nonn

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

Crossrefs

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

Programs

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 *)

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