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

1, 0, 1, 1, 16, 76, 1016, 10284, 157340, 2411756, 44953712, 899824256, 20283419872, 495216726096, 13202082981712, 378896535199888, 11690436112988224, 385173160930360192, 13509981115738946816, 502374681770910293568, 19746124320077115154112, 817908018939079281840320

Offset: 1

Author

Peter J. Cameron

Keywords

• nonn

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..200
• P. J. Cameron, Counting two-graphs related to trees, Elec. J. Combin., Vol. 2, #R4.

Crossrefs

Row sums are A361355.

Cf. A007831, A007832, A007833, A359986.

Programs

Mathematica

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 *)

PARI

\\ B(x) gives the e.g.f. of A359986.
B(n)={exp(2*x + intformal(serreverse(log(1 + x + O(x^n)) + log(1 + x + O(x^n)) - x)))}
seq(n)={Vec(serlaplace(log(subst(B(n), x, log(1 + x + O(x*x^n)))/(1 + x))))} \\ Andrew Howroyd, Oct 15 2024

Formula

E.g.f.: -2*LambertW(-1/2*exp(-1/2)*(1+x)^(1/2))/(1+x). - Vladeta Jovovic, Aug 21 2006

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

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

Extensions

a(20) onwards from Andrew Howroyd, Oct 15 2024