login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A006129 a(0), a(1), a(2), ... satisfy Sum_{k=0..n} a(k)*binomial(n,k) = 2^binomial(n,2), for n >= 0.
(Formerly M3678)
135
1, 0, 1, 4, 41, 768, 27449, 1887284, 252522481, 66376424160, 34509011894545, 35645504882731588, 73356937912127722841, 301275024444053951967648, 2471655539737552842139838345, 40527712706903544101000417059892, 1328579255614092968399503598175745633 (list; graph; refs; listen; history; text; internal format)
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
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
Row sums of A054548.
Cf. A322661 (if loops allowed), A086193 (directed edges), A002494 (unlabeled).
Sequence in context: A363302 A361182 A192547 * A244437 A265003 A193363
KEYWORD
nonn,nice,easy
AUTHOR
EXTENSIONS
More terms from Vladeta Jovovic, Apr 09 2000
STATUS
approved

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

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 28 08:22 EDT 2024. Contains 371236 sequences. (Running on oeis4.)