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%I #26 Mar 18 2018 15:43:59
%S 1,0,3,60,2835,219240,25519725,4169185020,910363278825,
%T 256123949281200,90240816705714675,38923077574032151500,
%U 20174526711617730727275,12373285262231460281715000,8863077725980930704895768125,7332455066541096999983523547500
%N The number of simple labeled graphs on 2n nodes whose connected components are even length cycles.
%H Robert Israel, <a href="/A202065/b202065.txt">Table of n, a(n) for n = 0..225</a>
%F E.g.f. for aerated sequence: exp(-x^2/4)/(1-x^2)^(1/4).
%F a(n) ~ (2*n)! * 2^(1/4)*exp(-1/4)*Gamma(3/4)/((2*n)^(3/4)*Pi). - _Vaclav Kotesovec_, Sep 24 2013
%F a(n) = ((2n)!/n!)*2F0(1/4,-n;;4)*(-1/4)^n. - _Benedict W. J. Irwin_, May 24 2016
%F (4n^3-n)a(n-1) + (4n^2+2n)a(n) - a(n+1) = 0. - _Robert Israel_, Mar 02 2017
%p f:= gfun:-rectoproc({(4*n^3-n)*a(n-1) + (4*n^2+2*n)*a(n) - a(n+1)=0,a(0)=1,a(1)=0},a(n),remember):
%p map(f, [$0..30]); # _Robert Israel_, Mar 02 2017
%t nn = 30; a = Log[1/(1 - x^2)^(1/4)] - x^2/4; Table[i, {i, 0, nn, 2}]! CoefficientList[Series[Exp[a], {x, 0, nn}], x][[Table[i, {i, 1, nn+1, 2}]]]
%t Table[((2 n)!/n!) HypergeometricPFQ[{1/4, -n}, {}, 4] (-1/4)^n, {n, 0, 15}] (* _Benedict W. J. Irwin_, May 24 2016 *)
%Y Cf. A001205, A053532, A053533.
%K nonn
%O 0,3
%A _Geoffrey Critzer_, Dec 10 2011
%E a(14) and e.g.f. corrected by _Robert Israel_, Mar 02 2017
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