A007107 Number of labeled 2-regular digraphs with n nodes.

1, 0, 0, 1, 9, 216, 7570, 357435, 22040361, 1721632024, 166261966956, 19459238879565, 2714812050902545, 445202898702992496, 84798391618743138414, 18567039007438379656471, 4631381194792101913679985, 1305719477625154539392776080, 413153055417968797025496881656

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• nice

Comments

Or number of n X n matrices with exactly two 1's in each row and column which are not in the main diagonal, other entries 0 (cf. A001499). - Vladimir Shevelev, Mar 22 2010

Number of 2-factors of the n-crown graph. - Andrew Howroyd, Feb 28 2016

References

• R. W. Robinson, personal communication.
• R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1982.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..254 (first 49 terms from R. W. Robinson)
• O. Gonzalez, C. Beltran and I. Santamaria, On the Number of Interference Alignment Solutions for the K-User MIMO Channel with Constant Coefficients, arXiv preprint arXiv:1301.6196 [cs.IT], 2013. - From _N. J. A. Sloane_, Feb 19 2013
• R. J. Mathar, OEIS A007107, Mar 15 2019

Crossrefs

Cf. column t=0 of A284989.

Cf. A007108 (log transform), A197458 (row and column sum <=2), A219889 (unlabeled), A001499 (loops allowed).

Programs

Maple

a:= proc(n) option remember; `if`(n<5, ((n-1)*(n-2)/2)^2,
      (n-1)*(2*(n^3-2*n^2+n+1)*a(n-1)/(n-2)+((n^2-2*n+2)*
      (n+1)*a(n-2) +(2*n^2-6*n+1)*n*a(n-3)+(n-3)*(a(n-4)*
      (n^3-5*n^2+3)-(n-4)*(n-1)*(n+1)*a(n-5))))/(2*n))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Apr 10 2017

Mathematica

Table[Sum[Sum[Sum[(-1)^(k+j-s)*n!*(n-k)!*(2n-k-2j-s)!/(s!*(k-s)!*(n-k-j)!^2*j!*2^(2n-2k-j)),{j,0,n-k}],{s,0,k}],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, May 09 2014 after Shanzhen Gao *)

PARI

a(n)=sum(k=0,n, sum(s=0,k, sum(j=0,n-k, (-1)^(k+j-s)*n!*(n-k)!*(2*n-k-2*j-s)!/(s!*(k-s)!*(n-k-j)!^2*j!*2^(2*n-2*k-j))))) \\ Charles R Greathouse IV, Feb 08 2017

Formula

a(n) = Sum_{k=0..n} Sum_{s=0..k} Sum_{j=0..n-k} (-1)^(k+j-s)*n!*(n-k)!*(2n-k-2j-s)!/(s!*(k-s)!*(n-k-j)!^2*j!*2^(2n-2k-j)). - Shanzhen Gao, Nov 05 2007

a(n) ~ 2*sqrt(Pi) * n^(2*n+1/2) / exp(2*n+5/2). - Vaclav Kotesovec, May 09 2014