A113235 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.

1, 1, 1, 7, 49, 301, 2281, 21211, 220417, 2528569, 32014801, 442974511, 6638604721, 107089487077, 1849731389689, 34051409587651, 665366551059841, 13751213558077681, 299644435399909537, 6864906328749052759, 164941239260973870001, 4146673091958686331421

Offset

0,4

Links

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

Formula

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... .

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

From Vaclav Kotesovec, Nov 13 2017: (Start)

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).

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

(End)

Maple

a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-j)*binomial(n-1, j-1)*j!, j=[1, $3..n]))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, May 10 2016

Mathematica

f[n_] := n!*Sum[(-1)^k*LaguerreL[n - 2*k, -1, -1]/k!, {k, 0, Floor[n/2]}]; Table[ f[n], {n, 0, 19}]
Range[0, 19]!*CoefficientList[ Series[ Exp[x*(1 - x + x^2)/(1 - x)], {x, 0, 19}], x] (* Robert G. Wilson v, Oct 21 2005 *)

Prog

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

Crossrefs

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

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

Sequence in context: A188748 A188986 A146884 * A294261 A294293 A357146

Adjacent sequences: A113232 A113233 A113234 * A113236 A113237 A113238

Keyword

nonn

Author

Karol A. Penson, Oct 19 2005

Status

approved