OFFSET
0,4
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).
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..200
H. Gupta, Enumeration of symmetric matrices, Duke Math. J., 35 (1968), vol 3, 653-659.
H. Gupta, Enumeration of symmetric matrices (annotated scanned copy)
FORMULA
It appears that e.g.f. = 1 + Integral_{t = 0..x} ((1-t)^(-3/2)*exp( t*(t^2+3*t-2)/(4-4*t) ). - Mark van Hoeij, Oct 25 2011
Recursion: a(n) = (n-1) a(n-1) + (n-2)^2 a(n-2) - (n-2)(n-3)(n-4) a(n-3) - (1/2) (n-2)(n-3)(n-4) a(n-4) - (1/2)(n-2)(n-3)(n-4)(n-5) a(n-5). - Robert Israel, Aug 05 2013
a(n) ~ exp(sqrt(2*n)-n-3/2) * n^(n-1/2) * (1+31/(24*sqrt(2*n))). - Vaclav Kotesovec, Aug 14 2013
EXAMPLE
a(3) = 3 because there are 3 symmetric 4 X 4 0-1 matrices with row sums 2 and first row 1 1 0 0, namely
1100, 1100, 1100,
1001, 1010, 1100,
0011, 0101, 0011,
0110, 0011, 0011.
MAPLE
a:= proc(n) a(n):= `if`(n<2, 1, (n-1) *a(n-1) +(n-2)^2 *a(n-2) -
(n-2)*(n-3)*(n-4)* a(n-3) - (1/2)* (n-2)*(n-3)*(n-4)* a(n-4) -
(1/2)*(n-2)*(n-3)*(n-4)*(n-5)* a(n-5))
end:
seq(a(n), n=0..30); # Robert Israel, Aug 05 2013
MATHEMATICA
max = 30; egf = 1 + Integrate[(1-t)^(-3/2)*Exp[t (t^2 + 3 t - 2)/(4 - 4 t)] + O[t]^max // Normal, t]; CoefficientList[egf, t]* Range[0, max]! (* Jean-François Alcover, Apr 06 2017, after Mark van Hoeij *)
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
Better name from and edited by Robert Israel, Aug 05 2013
STATUS
approved