A113235 - OEIS

%I #25 Sep 08 2022 08:45:23

%S 1,1,1,7,49,301,2281,21211,220417,2528569,32014801,442974511,

%T 6638604721,107089487077,1849731389689,34051409587651,665366551059841,

%U 13751213558077681,299644435399909537,6864906328749052759,164941239260973870001,4146673091958686331421

%N Number of partitions of {1,..,n} into any number of lists of size not equal to 2, where a list means an ordered subset, cf. A000262.

%H Alois P. Heinz, <a href="/A113235/b113235.txt">Table of n, a(n) for n = 0..444</a>

%F Expression as a sum involving generalized Laguerre polynomials, in Mathematica notation: a(n)=n!*Sum[(-1)^k*LaguerreL[n - 2*k, -1, -1]/k!, {k, 0, Floor[n/2]}], n=0, 1... .

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

%F From _Vaclav Kotesovec_, Nov 13 2017: (Start)

%F a(n) = (2*n - 1)*a(n-1) - (n-1)*n*a(n-2) + 4*(n-2)*(n-1)*a(n-3) - 2*(n-3)*(n-2)*(n-1)*a(n-4).

%F a(n) ~ exp(-3/2 + 2*sqrt(n) - n) * n^(n-1/4) / sqrt(2) * (1 + 91/(48*sqrt(n))).

%F (End)

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

%p       a(n-j)*binomial(n-1, j-1)*j!, j=[1, $3..n]))

%p     end:

%p seq(a(n), n=0..30);  # _Alois P. Heinz_, May 10 2016

%t f[n_] := n!*Sum[(-1)^k*LaguerreL[n - 2*k, -1, -1]/k!, {k, 0, Floor[n/2]}]; Table[ f[n], {n, 0, 19}]

%t Range[0, 19]!*CoefficientList[ Series[ Exp[x*(1 - x + x^2)/(1 - x)], {x, 0, 19}], x] (* _Robert G. Wilson v_, Oct 21 2005 *)

%o (PARI) m=30; v=concat([1,1,7,49], vector(m-4)); for(n=5, m, v[n]=(2*n-1)*v[n-1]-(n-1)*n*v[n-2]+4*(n-1)*(n-2)*v[n-3]-2*(n-1)*(n-2)*(n-3)*v[n -4]); concat([1], v) \\ _G. C. Greubel_, May 16 2018

%o (PARI) x='x+O('x^99); Vec(serlaplace(exp(x*(1-x+x^2)/(1-x)))) \\ _Altug Alkan_, May 17 2018

%o (Magma) I:=[1, 1, 7, 49]; [1] cat [n le 4 select I[n] else (2*n-1)*Self(n -1) - (n-1)*n*Self(n-2) +4*(n-1)*(n-2)*Self(n-3) -2*(n-1)*(n-2)*(n-3)* Self(n-4): n in [1..30]]; // _G. C. Greubel_, May 16 2018

%Y Cf. A000262, A000266, A005251, A052845, A097514.

%Y This sequence, A113236 and A113237 all describe the same type of mathematical structure: lists with some restrictions.

%K nonn

%O 0,4

%A _Karol A. Penson_, Oct 19 2005