A001866 Number of connected graphs with n nodes and n edges.

0, 0, 1, 24, 936, 56640, 4968000, 598328640, 94916183040, 19200422062080, 4826695329792000, 1476585999504000000, 540272647694971699200, 233019960215154829516800, 117009251702203840384204800, 67680314823703303654732800000, 44677678066673631080900198400000

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Or number of n X n (0,1) matrices with two 1's in each row the permanent of which equals to 2. Note that, if (0,1) matrix with two 1's in each row has positive permanent, then it is equal to a power of 2. - Vladimir Shevelev, Mar 25 2010

References

• V. S. Shevelev, On the permanent of the stochastic (0,1)-matrices with equal row sums, Izvestia Vuzov of the North-Caucasus region, Nature sciences 1 (1997), 21-38 (in Russian). - Vladimir Shevelev, Mar 25 2010
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n = 0..100
• T. L. Austin, R. E. Fagen, W. F. Penney, and John Riordan, The number of components in random linear graphs, Ann. Math. Statist 30 1959 747-754.
• J. Riordan, Letter to N. J. A. Sloane, Aug. 1970

Crossrefs

Cf. A174586.

Programs

Mathematica

Join[{0}, Table[(n!^2*n^(n - 1)/2)*Sum[n^(-k)/(n - k)!, {k, 2, n}], {n, 20}]] (* T. D. Noe, Aug 10 2012 *)

Formula

Explicit formula: a(n) = (n!^2*n^(n-1)/2)*Sum_{k=2..n} n^(-k)/(n-k)!; Recursion: a(2)=1, for n>=3, a(n) = n!*((n-1)!/2+Sum_{k=2..n-1} (-1)^(n+k+1)*k^(n-k)*binomial(n,k)*a(k)/k!). - Vladimir Shevelev, Mar 25 2010

a(n) ~ Pi * n^(2*n) / (2*exp(n)). - Vaclav Kotesovec, Nov 30 2017