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%I M2947 N1188 #51 Mar 15 2023 11:53:59
%S 1,1,1,3,13,70,462,3592,32056,322626,3611890,44491654,597714474,
%T 8693651092,136059119332,2279212812480,40681707637888,770631412413148,
%U 15438647456063004,326091322648369684,7241563996136849260,168657537987709667976,4110364564664358194536
%N Number of symmetric 0-1 (n+1) X (n+1) matrices with row sums 2 and first row starting 1,1 for n > 0, a(0)=1.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Alois P. Heinz, <a href="/A001495/b001495.txt">Table of n, a(n) for n = 0..200</a>
%H H. Gupta, <a href="http://dx.doi.org/10.1215/S0012-7094-68-03567-9">Enumeration of symmetric matrices</a>, Duke Math. J., 35 (1968), vol 3, 653-659.
%H H. Gupta, <a href="/A000085/a000085.pdf">Enumeration of symmetric matrices</a> (annotated scanned copy)
%F 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
%F 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
%F a(n) ~ exp(sqrt(2*n)-n-3/2) * n^(n-1/2) * (1+31/(24*sqrt(2*n))). - _Vaclav Kotesovec_, Aug 14 2013
%e 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
%e 1100, 1100, 1100,
%e 1001, 1010, 1100,
%e 0011, 0101, 0011,
%e 0110, 0011, 0011.
%p a:= proc(n) a(n):= `if`(n<2, 1, (n-1) *a(n-1) +(n-2)^2 *a(n-2) -
%p (n-2)*(n-3)*(n-4)* a(n-3) - (1/2)* (n-2)*(n-3)*(n-4)* a(n-4) -
%p (1/2)*(n-2)*(n-3)*(n-4)*(n-5)* a(n-5))
%p end:
%p seq(a(n), n=0..30); # _Robert Israel_, Aug 05 2013
%t 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_ *)
%K nonn
%O 0,4
%A _N. J. A. Sloane_
%E Better name from and edited by _Robert Israel_, Aug 05 2013
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