A113236 - OEIS

%I #25 May 09 2024 13:26:11

%S 1,1,3,7,49,321,2851,24823,256257,2887489,36759331,507010791,

%T 7597222513,122184356737,2106356007939,38693238713431,754792977928321,

%U 15572911248409473,338800604611562947,7749991799652960199,185934065196259734321,4667877395135551746241

%N 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.

%H Alois P. Heinz, <a href="/A113236/b113236.txt">Table of n, a(n) for n = 0..444</a>

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

%F 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....

%F a(n) ~ exp(-3/2+2*sqrt(n)-n)*n^(n-1/4)/sqrt(2). - _Vaclav Kotesovec_, Jun 22 2013

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

%p       `if`(j=3, 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 Range[0, 18]!*CoefficientList[ Series[ Exp[x*(1-x^2+x^3)/(1 - x)], {x, 0, 18}], x] (* _Zerinvary Lajos_, Mar 23 2007 *)

%t 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_ *)

%o (PARI) x='x+O('x^30); Vec(serlaplace(exp(x*(1-x^2+x^3)/(1-x)))) \\ _G. C. Greubel_, May 17 2018

%o (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

%Y Cf. A052845, A113235.

%K nonn

%O 0,3

%A _Karol A. Penson_, Oct 19 2005