A000986 Number of n X n symmetric matrices with (0,1) entries and all row sums 2.

1, 0, 1, 4, 18, 112, 820, 6912, 66178, 708256, 8372754, 108306280, 1521077404, 23041655136, 374385141832, 6493515450688, 119724090206940, 2337913445039488, 48195668439235612, 1045828865817825264, 23826258064972682776, 568556266922455167040

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

Comments

a(n) is the number of simple labeled graphs on n nodes with all vertices of degree 1 or 2.

From R. J. Mathar, Apr 07 2017: (Start)

These are the row sums of the following triangle which shows the number of symmetric n X n {0,1} matrices with row and column sums 2 refined for trace t, 0 <= t <= n:

0: 1

1: 0 0

2: 0 0 1

3: 1 0 3 0

4: 3 0 12 0 3

5: 12 0 70 0 30 0

6: 70 0 465 0 270 0 15

7: 465 0 3507 0 2625 0 315 0

See also A001205 for column t=0. (End)

References

• 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).
• R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 5.2.8.
• Herbert S. Wilf, Generatingfunctionology, p. 104.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..445 (first 101 terms from T. D. Noe)
• H. Gupta, Enumeration of symmetric matrices, Duke Math. J., 35 (1968), vol 3, 653-659.
• H. Gupta, Enumeration of symmetric matrices (annotated scanned copy)
• Zhonghua Tan, Shanzhen Gao, and H. Niederhausen, Enumeration of (0,1) matrices with constant row and column sums, Appl. Math. Chin. Univ. 21 (4) (2006) 479-486.

Crossrefs

Cf. A000985, A001205.

Programs

Maple

a:= proc(n) option remember;
       `if`(n<2, 1-n, add(binomial (n-1, k-1)
        *(k! +`if`(k>2, (k-1)!, 0))/2 *a(n-k), k=2..n))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Feb 24 2011

Mathematica

a=1/(2(1-x))-1/2-x/2; b=(Log[1/(1-x)]-x-x^2/2)/2;
Range[0, 20]! CoefficientList[Series[Exp[a + b], {x, 0, 20}], x]
(* Second program: *)
a[n_] := a[n] = If[n<2, 1-n, Sum[Binomial[n-1, k-1]*(k! + If[k>2, (k-1)!, 0])/2*a[n-k], {k, 2, n}]]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 20 2017, after Alois P. Heinz *)

Formula

E.g.f.: (1-x)^(-1/2)*exp(-x-x^2/4 + x/((2*(1-x)))).

Sum_{a_1=0..n} Sum_{c=0..min(a_1, n - a_1)} Sum_{b=0..floor((n - a_1 - c)/2)} ((-1)^((n - a_1 - 2b - c) + b)*n!*(2a_1)!) / (2^(n + a_1 - 2c)*a_1!*(n - a_1 - 2b - c)!*b!*(2c)!*(a_1 - c)!). - Shanzhen Gao, Jun 05 2009

Conjecture: 2*a(n) +2*(-2*n+1)*a(n-1) +2*(n^2-2*n-1)*a(n-2) -2*(n-2)*(n-4)*a(n-3) +(n-1)*(n-2)*(n-3)*a(n-4) -(n-2)*(n-3)*(n-4)*a(n-5)=0. - R. J. Mathar, Aug 04 2013

Recurrence: 2*a(n) = 4*(n-1)*a(n-1) - 2*(n-3)*(n-1)*a(n-2) - (n-3)*(n-2)*(n-1)*a(n-4). - Vaclav Kotesovec, Feb 13 2014

a(n) ~ n^n * exp(sqrt(2*n)-n-3/2) / sqrt(2) * (1 + 43/(24*sqrt(2*n))). - Vaclav Kotesovec, Feb 13 2014