OFFSET
0,2
COMMENTS
a(n) is the number of 2-colored labeled graphs (as in A047863) such that the black nodes are labeled with {1,2,...,k} where k, 0<=k<=n, is the number of black nodes and the white nodes are labeled with {k+1,k+2,...,n}. These graphs form the desired binomial poset (for the case q=2) in the "task left to the reader" in the Stanley reference below. - Geoffrey Critzer, May 31 2020
REFERENCES
R. P. Stanley, Enumerative Combinatorics, Volume I, Second Edition, Cambridge, 2012, Example 3.18.3 e, page 323.
LINKS
Robert Israel, Table of n, a(n) for n = 0..116
FORMULA
G.f.: A(x) = Sum_{n>=0} x^n/(1-2^n*x).
Let E(x) = Sum_{n>=0} x^n/2^C(n,2). Then E(x)^2 = Sum_{n>=0} a(n)*x^n/2^C(n,2). - Geoffrey Critzer, May 31 2020
a(n) ~ c * 2^(n^2/4), where c = EllipticTheta[3, 0, 1/2] = JacobiTheta3(0,1/2) = 2.128936827211877158669458548544951324612516539940878092889... if n is even and c = EllipticTheta[2, 0, 1/2] = JacobiTheta2(0,1/2) = 2.128931250513027558591613402575350180853805396958448940969... if n is odd. - Vaclav Kotesovec, Jun 28 2021
EXAMPLE
A(x) = 1/(1-x) + x/(1-2x) + x^2/(1-4x) + x^3/(1-8x) + ...
MAPLE
N:= 25:
G:= series(add(x^n/(1-2^n*x), n=0..N), x, N+1):
seq(coeff(G, x, n), n=0..N)); # Robert Israel, Dec 11 2018
MATHEMATICA
a[n_]:= Sum[2^((n-k)*k), {k, 0, n}]; Array[a, 20, 0] (* Amiram Eldar, Dec 12 2018 *)
PROG
(PARI) a(n)=sum(k=0, n, 2^((n-k)*k))
(Magma) [(&+[2^(k*(n-k)): k in [0..n]]): n in [0..20]]; // G. C. Greubel, Jun 28 2021
(Sage) [sum(2^(k*(n-k)) for k in (0..n)) for n in (0..20)] # G. C. Greubel, Jun 28 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Mar 12 2006
STATUS
approved