0,4

Or, number of labeled 2-regular relations of order n.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

H. Anand, V. C. Dumir and H. Gupta, A combinatorial distribution problem, Duke Math. J., 33 (1966), 757-769.

R. Bricard, L'Interm\'{e}diaire des Math\'{e}maticiens, 8 (1901), 312-313.

L. Carlitz, Enumeration of symmetric arrays, {\em Duke Math. J.}, {\bf 33} (1966), 771-782.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 236, P(n,2).

L. Erlebach and O. Ruehr, Problem 79-5, SIAM Review. Solution by D. E. Knuth. Reprinted in Problems in Applied Mathematices, ed. M. Klamkin, SIAM, 1990, p. 350.

J. T. Lewis, Maximal $L$-free subsets of a square-free array, {\em Congressus Numerantium}, {\bf 141} (1999), 151-155.

J. H. van Lint and R. M. Wilson, {\em A Course in Combinatorics}, (Cambridge University Press, Cambridge, 1992), pp. 152-153. [The second edition is said to be a better reference.]

R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Cor. 5.5.11 (b).

M. L. Stein and P. R. Stein, Enumeration of Stochastic Matrices with Integer Elements. Report LA-4434, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, Jun 1970.

R. W. Robinson, Table of n, a(n) for n = 0..48

Index entries for sequences related to binary matrices

M. E. Kuczma, 0-1-Matrices with Line-Sums Equal to 2, Am. Math. Month. 99 (1992) 959-961, E3419.

a(n) = (n! (n-1) Gamma(n-1/2) / Gamma(1/2) ) * 1F1[2-n; 3/2-n; -1/2] [Erlebach and Ruehr]. This representation is exact, asymptotic and convergent.

a(n) ~ 2 sqrt(Pi) n^(2n + 1/2) e^(-2n - 1/2) [Knuth]

a(n) = (1/2)*n*(n-1)^2 * ( (2*n-3)*a(n-2) + (n-2)^2*a(n-3) ) (from Anand et al.)

Sum_{n >= 0} a(n)*x^n/(n!)^2 = exp(-x/2)/sqrt(1-x); a(n) = n(n-1)/2 [ 2 a(n-1) + (n-1) a(n-2) ] (Bricard)

b_n = a_n/n! satisfies b_n = (n-1)(b_{n-1} + b_{n-2}/2); e.g.f. for {b_n} and for derangements (A000166) are related by D(x) = B(x)^2.

lim(n->infinity) sqrt(n)*a(n)/(n!)^2 = A096411 [Kuczma]. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Sep 21 2007

(PARI) a(n)=if(n<2, n==0, (n^2-n)*(a(n-1)+(n-1)/2*a(n-2)))

Cf. A000681, A053871, A123544 (connected relations).

Sequence in context: A138462 A002896 A004996 this_sequence A147630 A132467 A000680

Adjacent sequences: A001496 A001497 A001498 this_sequence A001500 A001501 A001502

nonn,nice,easy

N. J. A. Sloane (njas(AT)research.att.com).