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

0,3

Links

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

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

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 *)

Prog

(PARI) x='x+O('x^30); Vec(serlaplace(exp(x*(1-x^2+x^3)/(1-x)))) \\ G. C. Greubel, May 17 2018
(Magma) m:=25; R<x>:=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

Crossrefs

Cf. A052845, A113235.

Sequence in context: A118393 A362522 A113775 * A035499 A273092 A120788

Adjacent sequences: A113233 A113234 A113235 * A113237 A113238 A113239

Keyword

nonn

Author

Karol A. Penson, Oct 19 2005

Status

approved