OFFSET
0,4
COMMENTS
Also labeled graphs on n unisolated nodes (inverse binomial transform of A006125). - Vladeta Jovovic, Apr 09 2000
Also the number of edge covers of the complete graph K_n. - Eric W. Weisstein, Mar 30 2017
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..50
A. N. Bhavale, B. N. Waphare, Basic retracts and counting of lattices, Australasian J. of Combinatorics (2020) Vol. 78, No. 1, 73-99.
C. L. Mallows & N. J. A. Sloane, Emails, May 1991
N. J. A. Sloane, Transforms
R. Tauraso, Edge cover time for regular graphs, JIS 11 (2008) 08.4.4.
Eric Weisstein's World of Mathematics, Complete Graph
Eric Weisstein's World of Mathematics, Edge Cover
FORMULA
a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*2^binomial(k, 2).
E.g.f.: A(x)/exp(x) where A(x) = Sum_{n>=0} 2^C(n,2) x^n/n!. - Geoffrey Critzer, Oct 21 2011
a(n) ~ 2^(n*(n-1)/2). - Vaclav Kotesovec, May 04 2015
EXAMPLE
2^binomial(n,2) = 1 + binomial(n,2) + 4*binomial(n,3) + 41*binomial(n,4) + 768*binomial(n,5) + ...
MAPLE
a:= proc(n) option remember; `if`(n=0, 1,
2^binomial(n, 2) - add(a(k)*binomial(n, k), k=0..n-1))
end:
seq(a(n), n=0..20); # Alois P. Heinz, Oct 26 2012
MATHEMATICA
a = Sum[2^Binomial[n, 2] x^n/n!, {n, 0, 20}]; Range[0, 20]! CoefficientList[Series[a/Exp[x], {x, 0, 20}], x] (* Geoffrey Critzer, Oct 21 2011 *)
Table[Sum[(-1)^(n - k) Binomial[n, k] 2^Binomial[k, 2], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, May 04 2015 *)
PROG
(PARI) for(n=0, 25, print1(sum(k=0, n, (-1)^(n-k)*binomial(n, k)*2^binomial(k, 2)), ", ")) \\ G. C. Greubel, Mar 30 2017
(Python)
from sympy.core.cache import cacheit
from sympy import binomial
@cacheit
def a(n): return 1 if n==0 else 2**binomial(n, 2) - sum(a(k)*binomial(n, k) for k in range(n))
print([a(n) for n in range(21)]) # Indranil Ghosh, Aug 12 2017
CROSSREFS
KEYWORD
nonn,nice,easy
AUTHOR
EXTENSIONS
More terms from Vladeta Jovovic, Apr 09 2000
STATUS
approved