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%I #26 Mar 18 2022 00:10:26
%S 1,2,4,18,60,252,1860,9552,59616,565920,4051872,33805440,465239808,
%T 4294865664,35413136640,768372168960,8757710173440,79772814777600,
%U 1986906367584000,22082635812268800,280886415019776000,7683780010315046400
%N Number of partitions of {1, ..., 2n} into coprime pairs.
%H Seiichi Manyama, <a href="/A009679/b009679.txt">Table of n, a(n) for n = 1..30</a> calculated by Herman Jamke's code.
%H Carl Pomerance, <a href="https://arxiv.org/abs/2203.03085">Coprime permutations</a>, arXiv:2203.03085 [math.NT], 2022.
%F a(n) = sqrt(A005326(2n)). - _T. D. Noe_, Feb 10 2007
%F a(n) = permanent(m), where the n X n matrix m is defined by m(i,j) = 1 or 0, depending on whether gcd(2i,2j-1) is 1 or >1, respectively. - _T. D. Noe_, Feb 11 2007
%o (PARI) permRWNb(a)=n=matsize(a)[1]; if(n==1,return(a[1,1])); sg=1; nc=0; in=vectorv(n); x=in;x=a[,n]-sum(j=1,n,a[,j])/2; p=prod(i=1,n,x[i]); for(k=1,2^(n-1)-1,sg=-sg; j=valuation(k,2)+1;z=1-2*in[j]; in[j]+=z; nc+=z; x+=z*a[,j]; p+=prod(i=1,n,x[i],sg)); return(2*(2*(n%2)-1)*p)
%o for(n=1,26,a=matrix(n,n,i,j,gcd(2*i,2*j-1)==1);print1(permRWNb(a)", ")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), May 13 2007
%Y Cf. A001147 for the number of partitions (pairings) in unrestricted pairs.
%K nonn
%O 1,2
%A _David W. Wilson_
%E More terms from _T. D. Noe_, Feb 10 2007
%E More terms from _T. D. Noe_, Feb 11 2007
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