OFFSET
0,4
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.
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-François Alcover, Feb 20 2017, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,nice,easy
AUTHOR
STATUS
approved