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
Keywords
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..444
Crossrefs
Programs
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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
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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 *) -
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
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PARI
x='x+O('x^99); Vec(serlaplace(exp(x*(1-x+x^2)/(1-x)))) \\ Altug Alkan, May 17 2018
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)