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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. 7
1, 1, 1, 7, 49, 301, 2281, 21211, 220417, 2528569, 32014801, 442974511, 6638604721, 107089487077, 1849731389689, 34051409587651, 665366551059841, 13751213558077681, 299644435399909537, 6864906328749052759, 164941239260973870001, 4146673091958686331421 (list; graph; refs; listen; history; text; internal format)
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 A133046

Adjacent sequences:  A113232 A113233 A113234 * A113236 A113237 A113238

KEYWORD

nonn

AUTHOR

Karol A. Penson, Oct 19 2005

STATUS

approved

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Last modified April 17 22:19 EDT 2021. Contains 343071 sequences. (Running on oeis4.)