A007107 Number of labeled 2-regular digraphs with n nodes. (Formerly M4668)

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

Offset

0,5

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

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

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 *)

Prog

(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

Crossrefs

Cf. column t=0 of A284989.

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

Sequence in context: A067426 A250548 A007108 * A217042 A064633 A084942

Adjacent sequences: A007104 A007105 A007106 * A007108 A007109 A007110

Keyword

nonn,nice

Author

N. J. A. Sloane

Status

approved