Logo

Greetings from The On-Line Encyclopedia of Integer Sequences!

Hints

Search: id:A001818
Displaying 1-1 of 1 results found. page 1
     Format: long | short | internal | text      Sort: relevance | references | number      Highlight: on | off
A001818 Squares of double factorials: (1*3*5*...*(2n-1))^2 = ((2*n-1)!!)^2.
(Formerly M4669 N1997)
+0
16
1, 1, 9, 225, 11025, 893025, 108056025, 18261468225, 4108830350625, 1187451971330625, 428670161650355625, 189043541287806830625, 100004033341249813400625, 62502520838281133375390625 (list; graph; listen)
OFFSET

0,3

COMMENT

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.

REFERENCES

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

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

LINKS

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

Eric Weisstein's World of Mathematics, Struve function

FORMULA

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.

EXAMPLE

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.

PROGRAM

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

CROSSREFS

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

KEYWORD

nonn,easy,nice

AUTHOR

njas

page 1

Search completed in 0.002 seconds

Lookup | Welcome | Find friends | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Puzzles | Hot | Classics
More pages | Superseeker | Maintained by N. J. A. Sloane (njas@research.att.com)

Last modified May 16 01:24 EDT 2008. Contains 139630 sequences.


AT&T Labs Research