login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A047863 Number of labeled graphs with 2-colored nodes where black nodes are only connected to white nodes and vice versa. 33
1, 2, 6, 26, 162, 1442, 18306, 330626, 8488962, 309465602, 16011372546, 1174870185986, 122233833963522, 18023122242478082, 3765668654914699266, 1114515608405262434306, 467221312005126294077442, 277362415313453291571118082 (list; graph; refs; listen; history; text; internal format)
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

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.

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 *)

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 xrange(n + 1)]) # Indranil Ghosh, Jun 03 2017

CROSSREFS

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified February 20 10:56 EST 2018. Contains 299385 sequences. (Running on oeis4.)