A135079 E.g.f. A(x) = Sum_{n>=0} exp(3^n*x)*x^n/n!.

1, 2, 8, 56, 704, 15392, 593408, 39691136, 4650143744, 944100803072, 334651494268928, 205435333440321536, 219775256161359233024, 407034554694060677537792, 1312205966809501720566038528

Offset

0,2

Comments

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

Links

G. C. Greubel, Table of n, a(n) for n = 0..75

Formula

a(n) = Sum_{k=0..n} C(n, k)*3^(k*(n-k)).

O.g.f.: A(x) = Sum_{n>=0} x^n/(1 - 3^n*x)^(n+1). - Paul D. Hanna, Aug 08 2009

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

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

Mathematica

Table[Sum[Binomial[n, k]*3^(k*(n-k)), {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jun 24 2013 *)

Prog

(PARI) {a(n)=sum(k=0, n, binomial(n, k)*3^(k*(n-k)))}
(PARI) /* E.g.f.: */ {a(n)=n!*polcoeff(sum(k=0, n, exp(3^k*x +x*O(x^n))*x^k/k!), n)}
(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

Crossrefs

Cf. A047863 (variant). A188457.

Sequence in context: A203199 A348875 A302999 * A084872 A254231 A191713

Adjacent sequences: A135076 A135077 A135078 * A135080 A135081 A135082

Keyword

nonn,easy

Author

Paul D. Hanna, Nov 24 2007

Status

approved