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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.

%I A113236 #25 May 09 2024 13:26:11
%S A113236 1,1,3,7,49,321,2851,24823,256257,2887489,36759331,507010791,
%T A113236 7597222513,122184356737,2106356007939,38693238713431,754792977928321,
%U A113236 15572911248409473,338800604611562947,7749991799652960199,185934065196259734321,4667877395135551746241
%N 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.
%H A113236 Alois P. Heinz, <a href="/A113236/b113236.txt">Table of n, a(n) for n = 0..444</a>
%F A113236 E.g.f.: exp(x*(1-x^2+x^3)/(1-x)).
%F A113236 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 A113236 a(n) ~ exp(-3/2+2*sqrt(n)-n)*n^(n-1/4)/sqrt(2). - _Vaclav Kotesovec_, Jun 22 2013
%p A113236 a:= proc(n) option remember; `if`(n=0, 1, add(
%p A113236       `if`(j=3, 0, a(n-j)*binomial(n-1, j-1)*j!), j=1..n))
%p A113236     end:
%p A113236 seq(a(n), n=0..30);  # _Alois P. Heinz_, May 10 2016
%t A113236 Range[0, 18]!*CoefficientList[ Series[ Exp[x*(1-x^2+x^3)/(1 - x)], {x, 0, 18}], x] (* _Zerinvary Lajos_, Mar 23 2007 *)
%t A113236 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 A113236 (PARI) x='x+O('x^30); Vec(serlaplace(exp(x*(1-x^2+x^3)/(1-x)))) \\ _G. C. Greubel_, May 17 2018
%o A113236 (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 A113236 Cf. A052845, A113235.
%K A113236 nonn
%O A113236 0,3
%A A113236 _Karol A. Penson_, Oct 19 2005