0,3

Number of permutations in S_{2n} in which all cycles have even length (cf. A087137).

a(n)=(2*n-1)!*sum(binomial(2*k,k)/4^k,k=0..n-1), n>=1. W. Lang Aug 23 2005 (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de)

Also number of permutations in S_{2n} in which all cycles have odd length. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Aug 10 2007

a(n)=sum over all multinomials M2(2*n,k), k from {1..p(2*n)} restricted to partitions with only even parts. p(2*n)= A000041(2*n) (partition numbers), and for the M2-multinomial numbers in A-St order see A036039(2*n,k). W. Lang, Aug 07 2007.

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.34(c).

T. D. Noe, Table of n, a(n) for n=0..50

Eric Weisstein's World of Mathematics, Struve function

a(0)=1, a(n)=(2*n-1)^2*a(n-1), n>0.

a(n) ~ 2*2^(2*n)*e^(-2*n)*n^(2*n) - Joe Keane (jgk(AT)jgk.org), Jun 06 2002

For even n, a(n) = n!-((n/2)!!)^2. - Yuval Dekel, Oct 31, 2001

E.g.f.: 1/sqrt(1-x^2) = Sum_{n >= 0} a(n)*x^(2*n)/(2*n)!. Also arcsin(x) = Sum_{n >= 0} a(n)*x^(2*n+1)/(2*n+1)!. - Michael Somos Jul 03 2002

(-1)^n*a(n) is the coefficient of x^0 in prod(k=1, 2*n, x+2*k-2*n-1). - Benoit Cloitre and Michael Somos, Nov 22, 2002.

Multinomial representation for a(2): partitions of 2*2=4 with even parts only: (4) with position k=1, (2^2) with k=3; M2(4,1)= 6 and M2(4,3)= 3, adding up to a(2)=9.

(PARI) a(n)=((2*n)!/(n!*2^n))^2

A001818(n)=A001147(n)^2. Cf. A002454.

Bisection of A012248.

Right-hand column 1 in triangle A008956.

a(n)= A111595(2*n, 0).

Adjacent sequences: A001815 A001816 A001817 this_sequence A001819 A001820 A001821

Sequence in context: A012749 A079727 A128492 this_sequence A095363 A085799 A075127

nonn,easy,nice

njas