A047863 Number of labeled graphs with 2-colored nodes where black nodes are only connected to white nodes and vice versa.

1, 2, 6, 26, 162, 1442, 18306, 330626, 8488962, 309465602, 16011372546, 1174870185986, 122233833963522, 18023122242478082, 3765668654914699266, 1114515608405262434306, 467221312005126294077442, 277362415313453291571118082, 233150477220213193598856331266

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• nice

Comments

Row sums of A111636. - Peter Bala, Sep 30 2012

Column 2 of Table 2 in Read. - Peter Bala, Apr 11 2013

It appears that 5 does not divide a(n), that a(n) is even for n>0, that 3 divides a(2n) for n>0, that 7 divides a(6n+5), and that 13 divides a(12n+3). - Ralf Stephan, May 18 2013

Examples

			For n=2, {1,2 black, not connected}, {1,2 white, not connected}, {1 black, 2 white, not connected}, {1 black, 2 white, connected}, {1 white, 2 black, not connected}, {1 white, 2 black, connected}.
G.f. = 1 + 2*x + 6*x^2 + 26*x^3 + 162*x^4 + 1442*x^5 + 18306*x^6 + ...
		

References

• H. S. Wilf, Generatingfunctionology, Academic Press, NY, 1990, p. 79, Eq. 3.11.2.

Links

• T. D. Noe, Table of n, a(n) for n = 0..50
• P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
• S. R. Finch, Bipartite, k-colorable and k-colored graphs
• S. R. Finch, Bipartite, k-colorable and k-colored graphs, June 5, 2003. [Cached copy, with permission of the author]
• A. Gainer-Dewar and I. M. Gessel, Enumeration of bipartite graphs and bipartite blocks, arXiv:1304.0139 [math.CO], 2013.
• D. A. Klarner, The number of graded partially ordered sets, J. Combin. Theory, 6 (1969), 12-19. [Annotated scanned copy]
• Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
• R. C. Read, The number of k-colored graphs on labelled nodes, Canad. J. Math., 12 (1960), 410-414.
• R. P. Stanley, Acyclic orientation of graphs Discrete Math. 5 (1973), 171-178. North Holland Publishing Company.
• Martin Svatoš, Peter Jung, Jan Tóth, Yuyi Wang, and Ondřej Kuželka, On Discovering Interesting Combinatorial Integer Sequences, arXiv:2302.04606 [cs.LO], 2023, p. 17.
• Eric Weisstein's World of Mathematics, k-Colorable Graph
• H. S. Wilf, Generatingfunctionology, 2nd edn., Academic Press, NY, 1994, p. 88, Eq. 3.11.2.

Crossrefs

Column k=2 of A322280.

Cf. A135079 (variant).

Cf. A000683, A001831, A002031, A033995, A047864, A052332, A111636, A191371, A223887.

Programs

Magma

A047863:= func< n | (&+[Binomial(n,k)*2^(k*(n-k)): k in [0..n]]) >;
[A047863(n): n in [0..40]]; // G. C. Greubel, Nov 03 2024

Mathematica

Table[Sum[Binomial[n,k]2^(k(n-k)),{k,0,n}],{n,0,20}] (* Harvey P. Dale, May 09 2012 *)
nmax = 20; CoefficientList[Series[Sum[E^(2^k*x)*x^k/k!, {k, 0, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jun 05 2019 *)

PARI

{a(n)=n!*polcoeff(sum(k=0,n,exp(2^k*x +x*O(x^n))*x^k/k!),n)} \\ Paul D. Hanna, Nov 27 2007

PARI

{a(n)=polcoeff(sum(k=0, n, x^k/(1-2^k*x +x*O(x^n))^(k+1)), n)} \\ Paul D. Hanna, Mar 08 2008

PARI

N=66; x='x+O('x^N); egf = sum(n=0, N, exp(2^n*x)*x^n/n!);
Vec(serlaplace(egf))  \\ Joerg Arndt, May 04 2013

Python

from sympy import binomial
def a(n): return sum([binomial(n, k)*2**(k*(n - k)) for k in range(n + 1)]) # Indranil Ghosh, Jun 03 2017

SageMath

def A047863(n): return sum(binomial(n,k)*2^(k*(n-k)) for k in range(n+1))
[A047863(n) for n in range(41)] # G. C. Greubel, Nov 03 2024

Formula

a(n) = Sum_{k=0..n} binomial(n, k)*2^(k*(n-k)).

a(n) = 4 * A000683(n) + 2. - Vladeta Jovovic, Feb 02 2000

E.g.f.: Sum_{n>=0} exp(2^n*x)*x^n/n!. - Paul D. Hanna, Nov 27 2007

O.g.f.: Sum_{n>=0} x^n/(1 - 2^n*x)^(n+1). - Paul D. Hanna, Mar 08 2008

From Peter Bala, Apr 11 2013: (Start)

Let E(x) = Sum_{n >= 0} x^n/(n!*2^C(n,2)) = 1 + x + x^2/(2!*2) + x^3/(3!*2^3) + .... Then a generating function is E(x)^2 = 1 + 2*x + 6*x^2/(2!*2) + 26*x^3/(3!*2^3) + .... In general, E(x)^k, k = 1, 2, ..., is a generating function for labeled k-colored graphs (see Stanley). For other examples see A191371 (k = 3) and A223887 (k = 4).

If A(x) = 1 + 2*x + 6*x^2/2! + 26*x^3/3! + ... denotes the e.g.f. for this sequence then sqrt(A(x)) = 1 + x + 2*x^2/2! + 7*x^3/3! + ... is the e.g.f. for A047864, which counts labeled 2-colorable graphs. (End)

a(n) ~ c * 2^(n^2/4+n+1/2)/sqrt(Pi*n), where c = Sum_{k = -infinity..infinity} 2^(-k^2) = EllipticTheta[3, 0, 1/2] = 2.128936827211877... if n is even and c = Sum_{k = -infinity..infinity} 2^(-(k+1/2)^2) = EllipticTheta[2, 0, 1/2] = 2.12893125051302... if n is odd. - Vaclav Kotesovec, Jun 24 2013

Extensions

Better description from Christian G. Bower, Dec 15 1999