A108704 Number of partitions of 112233...nn into n pairs.

1, 1, 4, 18, 126, 1110, 12120, 156660, 2341500, 39701340, 752839920, 15785181720, 362606123880, 9055825538760, 244296192460320, 7079382509799600, 219321853964413200, 7233629128601475600, 253054306933115688000, 9358989706213886138400, 364860828050107348159200

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

Cf. A002135.

Sequence in context: A215691 A359460 A073511 * A001423 A308351 A291335

Adjacent sequences: A108701 A108702 A108703 * A108705 A108706 A108707

Keyword

nonn

Author

Miklos Kristof, Jun 20 2005

Status

approved