This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A135079 #18 Apr 20 2020 15:09:11
%S A135079 1,2,8,56,704,15392,593408,39691136,4650143744,944100803072,
%T A135079 334651494268928,205435333440321536,219775256161359233024,
%U A135079 407034554694060677537792,1312205966809501720566038528
%N A135079 E.g.f. A(x) = Sum_{n>=0} exp(3^n*x)*x^n/n!.
%C A135079 a(n) is the number of labeled graphs with (at most) 2 colors of vertices where vertices of the same color are never adjacent and the graphs may have up to 2 types of edges. - _Geoffrey Critzer_, Apr 20 2020
%H A135079 G. C. Greubel, <a href="/A135079/b135079.txt">Table of n, a(n) for n = 0..75</a>
%F A135079 a(n) = Sum_{k=0..n} C(n, k)*3^(k*(n-k)).
%F A135079 O.g.f.: A(x) = Sum_{n>=0} x^n/(1 - 3^n*x)^(n+1). - _Paul D. Hanna_, Aug 08 2009
%F A135079 Let E(x) = sum {n >= 0} x^n/(n!*3^C(n,2)). Then a generating function for this sequence is E(x)^2 = sum {n >= 0} a(n)*x^n/(n!*3^C(n,2)) = 1 + 2*x + 8*x^2/(2!*3) + 56*x^3/(3!*3^3) + 704*x^4/(4!*3^6) + .... Cf. A188457. - _Peter Bala_, Apr 01 2013
%F A135079 a(n) ~ c * 3^(n^2/4)*2^(n+1/2)/sqrt(Pi*n), where c = Sum_{k = -infinity..infinity} 3^(-k^2) = 1.6914596816817... if n is even and c = Sum_{k = -infinity..infinity} 3^(-(k+1/2)^2) = 1.69061120307521... if n is odd. - _Vaclav Kotesovec_, Jun 24 2013
%t A135079 Table[Sum[Binomial[n,k]*3^(k*(n-k)),{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Jun 24 2013 *)
%o A135079 (PARI) {a(n)=sum(k=0,n,binomial(n,k)*3^(k*(n-k)))}
%o A135079 (PARI) /* E.g.f.: */ {a(n)=n!*polcoeff(sum(k=0,n,exp(3^k*x +x*O(x^n))*x^k/k!),n)}
%o A135079 (PARI) {a(n)=polcoeff(sum(k=0,n,x^k/(1 - 3^k*x +x*O(x^n))^(k+1)),n)} \\ _Paul D. Hanna_, Aug 08 2009
%Y A135079 Cf. A047863 (variant). A188457.
%K A135079 nonn,easy
%O A135079 0,2
%A A135079 _Paul D. Hanna_, Nov 24 2007