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A007107
Number of labeled 2-regular digraphs with n nodes.
(Formerly M4668)
15
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
KEYWORD
nonn,nice
STATUS
approved