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A007834 Number of point labeled reduced 5-free two-graphs with n nodes.

%I A007834 #18 Oct 15 2024 18:07:03
%S A007834 1,0,1,1,16,76,1016,10284,157340,2411756,44953712,899824256,
%T A007834 20283419872,495216726096,13202082981712,378896535199888,
%U A007834 11690436112988224,385173160930360192,13509981115738946816,502374681770910293568,19746124320077115154112,817908018939079281840320
%N A007834 Number of point labeled reduced 5-free two-graphs with n nodes.
%H A007834 Andrew Howroyd, <a href="/A007834/b007834.txt">Table of n, a(n) for n = 1..200</a>
%H A007834 P. J. Cameron, <a href="http://www.combinatorics.org/Volume_2/volume2.html#R4">Counting two-graphs related to trees</a>, Elec. J. Combin., Vol. 2, #R4.
%F A007834 E.g.f.: -2*LambertW(-1/2*exp(-1/2)*(1+x)^(1/2))/(1+x). - _Vladeta Jovovic_, Aug 21 2006
%F A007834 a(n) ~ sqrt(2)*sqrt(4-exp(1)) * n^(n-1) / (8*exp(n-1)*(4*exp(-1)-1)^n). - _Vaclav Kotesovec_, Sep 30 2013
%F A007834 E.g.f.: log(B(log(1 + x))/(1 + x)), where B(x) is the e.g.f. of A359986. - _Andrew Howroyd_, Oct 15 2024
%t A007834 CoefficientList[Series[-2*LambertW[-1/2*E^(-1/2)*(1+x)^(1/2)]/(1+x), {x, 0, 15}], x]* Range[0, 15]! (* _Vaclav Kotesovec_, Sep 30 2013 *)
%o A007834 (PARI) \\ B(x) gives the e.g.f. of A359986.
%o A007834 B(n)={exp(2*x + intformal(serreverse(log(1 + x + O(x^n)) + log(1 + x + O(x^n)) - x)))}
%o A007834 seq(n)={Vec(serlaplace(log(subst(B(n), x, log(1 + x + O(x*x^n)))/(1 + x))))} \\ _Andrew Howroyd_, Oct 15 2024
%Y A007834 Row sums are A361355.
%Y A007834 Cf. A007831, A007832, A007833, A359986.
%K A007834 nonn
%O A007834 1,5
%A A007834 _Peter J. Cameron_
%E A007834 a(20) onwards from _Andrew Howroyd_, Oct 15 2024