A113236 Number of partitions of {1,..,n} into any number of lists of size not equal to 3, where a list means an ordered subset, cf. A000262.

1, 1, 3, 7, 49, 321, 2851, 24823, 256257, 2887489, 36759331, 507010791, 7597222513, 122184356737, 2106356007939, 38693238713431, 754792977928321, 15572911248409473, 338800604611562947, 7749991799652960199, 185934065196259734321, 4667877395135551746241

Offset: 0

Author

Karol A. Penson, Oct 19 2005

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..444

Crossrefs

Cf. A052845, A113235.

Programs

Magma

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(x*(1-x^2+x^3)/(1-x)))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 17 2018

Maple

a:= proc(n) option remember; `if`(n=0, 1, add(
      `if`(j=3, 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

Range[0, 18]!*CoefficientList[ Series[ Exp[x*(1-x^2+x^3)/(1 - x)], {x, 0, 18}], x] (* Zerinvary Lajos, Mar 23 2007 *)
a[n_] := a[n] = If[n==0, 1, Sum[If[j==3, 0, a[n-j]*Binomial[n-1, j-1]*j!], {j, 1, n}]]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 11 2017, after Alois P. Heinz *)

PARI

x='x+O('x^30); Vec(serlaplace(exp(x*(1-x^2+x^3)/(1-x)))) \\ G. C. Greubel, May 17 2018

Formula

E.g.f.: exp(x*(1-x^2+x^3)/(1-x)).

Expression as a sum involving generalized Laguerre polynomials, in Mathematica notation: a(n)=n!*Sum[(-1)^k*LaguerreL[n - 3*k, -1, -1]/k!, {k, 0, Floor[n/3]}], n=0, 1....

a(n) ~ exp(-3/2+2*sqrt(n)-n)*n^(n-1/4)/sqrt(2). - Vaclav Kotesovec, Jun 22 2013