A174586 -id:A174586 - cp's OEIS frontend

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

N. J. A. Sloane

nonn

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

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).

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

Cf. A174586.

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 *)

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

A174637 Number of n X n (0,1) matrices with two 1's in each row the permanent of which equals to 4.

Original entry on oeis.org

0, 0, 0, 18, 2400, 325800, 52496640, 10304300160, 2458401684480, 705918026419200, 241147866161664000, 96890287539173990400, 45304089884519168102400, 24415719893124157985587200, 15035096121857624246353920000, 10496828397482345253454479360000, 8250414679239607850470753370112000

Offset: 1

Vladimir Shevelev, Mar 25 2010

nonn

If a (0,1) matrix with two 1's in each row has positive permanent, then it equals to a power of 2.

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).

Max Alekseyev, Table of n, a(n) for n = 1..100

Cf. A000276, A001866, A174638, A174586, A377246.

a174637(n) = n!*(n-1)!/4 * sum(l=0,n-4, n^l/l! * sum(i=2, n-l-2, 1/i)); \\ Max Alekseyev, Oct 21 2024

a(n) = n!/4 * Sum_{l=0..n-4} binomial(n-1,l) * n^l * A000276(n-l). - Max Alekseyev, Oct 21 2024

G.f. for 4*a(n)/n!/(n-1)!: (W(-x)-ln(1+W(-x)))*(W(-x)/(1+W(-x)))^2, where W() is Lambert W-function. - Max Alekseyev, Oct 21 2024

Incorrect formula moved to A377246 and terms a(15) onward added by Max Alekseyev, Oct 21 2024

A174638 Number of n X n (0,1)-matrices with two 1's in each row and permanent equal to 8.

Original entry on oeis.org

0, 0, 0, 0, 0, 1350, 529200, 172872000, 58352555520, 21677788944000, 9059008787136000, 4286753834515891200, 2297335836334687948800, 1390520517156693315993600, 946759961227258909995264000

Offset: 1

Vladimir Shevelev, Mar 25 2010

nonn

If a (0,1)-matrix with two 1's in each row has positive permanent, then it equals to a power of 2.

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1967, Ch.4, 66-79.
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)

Cf. A000276, A001866, A174637, A174586.

For m >= 1, the number of n X n (0,1)-matrices with two 1's in each row with the permanent 2^m is n! * n^(n-1) / 2^m * Sum{k=2,...,n} k * n^(-k) * binomial(n,k) * d(k,m), where d(k,m) are associated Stirling numbers of the first kind (see Riordan, p. 75).

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

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A174637 Number of n X n (0,1) matrices with two 1's in each row the permanent of which equals to 4.

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A174638 Number of n X n (0,1)-matrices with two 1's in each row and permanent equal to 8.

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