A009679 Number of partitions of {1, ..., 2n} into coprime pairs.

1, 2, 4, 18, 60, 252, 1860, 9552, 59616, 565920, 4051872, 33805440, 465239808, 4294865664, 35413136640, 768372168960, 8757710173440, 79772814777600, 1986906367584000, 22082635812268800, 280886415019776000, 7683780010315046400

Offset

1,2

Links

Seiichi Manyama, Table of n, a(n) for n = 1..30 calculated by Herman Jamke's code.

Carl Pomerance, Coprime permutations, arXiv:2203.03085 [math.NT], 2022.

Formula

a(n) = sqrt(A005326(2n)). - T. D. Noe, Feb 10 2007

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

Prog

(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)
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

Crossrefs

Cf. A001147 for the number of partitions (pairings) in unrestricted pairs.

Sequence in context: A083694 A179040 A241685 * A007727 A303352 A226011

Adjacent sequences: A009676 A009677 A009678 * A009680 A009681 A009682

Keyword

nonn

Author

David W. Wilson

Extensions

More terms from T. D. Noe, Feb 10 2007

More terms from T. D. Noe, Feb 11 2007

Status

approved