%I A001818 M4669 N1997
%S A001818 1,1,9,225,11025,893025,108056025,18261468225,4108830350625,
%T A001818 1187451971330625,428670161650355625,189043541287806830625,
%U A001818 100004033341249813400625,62502520838281133375390625
%N A001818 Squares of double factorials: (1*3*5*...*(2n-1))^2 = ((2*n-1)!!)^2.
%C A001818 Number of permutations in S_{2n} in which all cycles have even length 
               (cf. A087137).
%C A001818 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)
%C A001818 Also number of permutations in S_{2n} in which all cycles have odd length. 
               - Vladeta Jovovic (vladeta(AT)eunet.rs), Aug 10 2007
%C A001818 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.
%C A001818 arcsinh(x) = sum((-1)^(n-1)*a(n)*x^(2*n-1)/(2*n-1)!, n=1..infinity) [From 
               James Buddenhagen (jbuddenh(AT)gmail.com), Mar 24 2009]
%D A001818 J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.
%D A001818 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001818 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001818 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see 
               Problem 5.34(c).
%H A001818 T. D. Noe, <a href="b001818.txt">Table of n, a(n) for n=0..50</a>
%H A001818 IBM, <a href="http://domino.research.ibm.com/Comm/wwwr_ponder.nsf/solutions/
               June2009.html">"Ponder This" puzzle for June, 2009.</a> [From Vladeta 
               Jovovic (vladeta(AT)eunet.yu), Jul 26 2009]
%H A001818 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Struvefunction.html">Struve function</a>
%F A001818 Contribution from Karol A.Penson (penson(AT)lptl.jussieu.fr), Oct 21 
               2009: (Start)
%F A001818 G.f.:sum(a(n)*x^n/(n!)^2,n=0..infinity)=2*EllipticK(2*sqrt(x))/Pi.
%F A001818 Asymptotically: a(n)=(2/((exp(-1/2))^2*(exp(1/2))^2)-1/(6*(exp(-1/2))^2*(exp(1/
               2))^2*n)+1/(144*(exp(-1/2))^2*(exp(1/2))^2*n^2)+O(1/n^3))*(2^n)^2/
               (((1/n)^n)^2*(exp(n))^2), n->infinity.
%F A001818 Integral representation as n-th moment of a positive function on a positive
%F A001818 halfaxis (solution of the Stieltjes moment problem), in Maple notation:
%F A001818 a(n)=int(x^n*BesselK(0,sqrt(x))/(Pi*sqrt(x)),x=0..infinity), n=0,1... 
               .
%F A001818 This solution is unique.
%F A001818 (End)
%F A001818 a(0)=1, a(n)=(2*n-1)^2*a(n-1), n>0.
%F A001818 a(n) ~ 2*2^(2*n)*e^(-2*n)*n^(2*n) - Joe Keane (jgk(AT)jgk.org), Jun 06 
               2002
%F A001818 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
%F A001818 (-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.
%F A001818 -arccos(x)+ pi/2 = x + x^3/3! + 9 x^5/5! + 225 x^7/7! + 11205 x^9/9! 
               + ... [From Tom Copeland (tcjpn(AT)msn.com), Oct 23 2008]
%e A001818 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.
%p A001818 a := proc(m) local k; 4^m*mul((-1)^k*(k-m-1/2),k=1..2*m) end; [From Peter 
               Luschny (peter(AT)luschny.de), Jun 01 2009]
%o A001818 f:=n->((2*n)!/(n!*2^n))^2;
%o A001818 (PARI) a(n)=((2*n)!/(n!*2^n))^2
%Y A001818 A001818(n)=A001147(n)^2. Cf. A002454.
%Y A001818 Bisection of A012248.
%Y A001818 Right-hand column 1 in triangle A008956.
%Y A001818 a(n)= A111595(2*n, 0).
%Y A001818 Sequence in context: A012749 A079727 A128492 this_sequence A095363 A138564 
               A158728
%Y A001818 Adjacent sequences: A001815 A001816 A001817 this_sequence A001819 A001820 
               A001821
%K A001818 nonn,easy,nice
%O A001818 0,3
%A A001818 N. J. A. Sloane (njas(AT)research.att.com).
%E A001818 Incorrect formula deleted by N. J. A. Sloane, Jul 03 2009