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A000986 Number of n X n symmetric matrices with (0,1) entries and all row sums 2.
(Formerly M3548 N1437)
10
1, 0, 1, 4, 18, 112, 820, 6912, 66178, 708256, 8372754, 108306280, 1521077404, 23041655136, 374385141832, 6493515450688, 119724090206940, 2337913445039488, 48195668439235612, 1045828865817825264, 23826258064972682776, 568556266922455167040 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

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

Comment 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

T. D. Noe, Table of n, a(n) for n = 0..100

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, H. Niederhausen, Enumeration of (0,1) matrices with constant row and column sums, Appl. Math. - A Journal of chin. Univ. 21 (4) (2006) 479-486.

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)!}$

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

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-Fran├žois Alcover, Feb 20 2017, after Alois P. Heinz *)

CROSSREFS

Cf. A000985, A001205.

Sequence in context: A060223 A144085 A003708 * A143920 A233534 A113356

Adjacent sequences:  A000983 A000984 A000985 * A000987 A000988 A000989

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified August 24 02:32 EDT 2017. Contains 291052 sequences.