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

Offset

0,2

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

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.

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

Example

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 + ...

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 *)

Prog

(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

Crossrefs

Column k=2 of A322280.

Cf. A001831, A002031, A052332.

Cf. A135079 (variant).

Cf. A111636. A033995, A047864, A191371, A223887.

Sequence in context: A135922 A213430 A103367 * A180349 A141713 A005272

Adjacent sequences: A047860 A047861 A047862 * A047864 A047865 A047866

Keyword

nonn,nice

Author

N. J. A. Sloane

Extensions

Better description from Christian G. Bower, Dec 15 1999

Status

approved