OFFSET
0,3
REFERENCES
Laszlo Lovasz, Combinatorial Problems and Solutions, AMS Chelsea Publishing, American Mathematical Society.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..400
FORMULA
E.g.f.: exp(x*x/2)/sqrt(1-2*x).
a(n) ~ 2^(n+1/2)*n^n/exp(n-1/8). - Vaclav Kotesovec, Sep 26 2013
a(n) = 2^n*(n-1/2)!*2F2((1-n)/2,-n/2;1/4 -n/2,3/4 - n/2; 1/8)/sqrt(Pi). - Benedict W. J. Irwin, May 25 2016
Conjecture: a(n)-(2*n-1)*a(n-1)-(n-1)*a(n-2)+2*(n-1)*(n-2)*a(n-3)=0. - R. J. Mathar, Jun 08 2016
From Emanuele Munarini, May 25 2022: (Start)
The exponential generating series A(t) satisfies the differential equation (1-2*t)*A'(t) = (1+t-2*t^2)*A(t), which is equivalent to the conjectured recurrence.
a(n) = Sum_{k=0..n/2} binomial(n,k)*binomial(2*n-4*k,n-2*k)*(n-k)!/2^(n-k).
(End)
EXAMPLE
Partitions of 1122 into 2 pairs: 11 22, 12 12, 12 21, 21 21 = 4 partitions so a(2)=4.
MAPLE
a:= n-> n! *coeff(series(exp(x*x/2)/sqrt(1-2*x), x, n+1), x, n):
seq (a(n), n=0..20);
MATHEMATICA
CoefficientList[Series[E^(x*x/2)/Sqrt[1-2*x], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Sep 26 2013 *)
PROG
(PARI) my(x='x+O('x^50)); Vec(serlaplace(exp(x*x/2)/sqrt(1-2*x))) \\ G. C. Greubel, May 24 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Miklos Kristof, Jun 20 2005
STATUS
approved