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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)
16
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

Alois P. Heinz, Table of n, a(n) for n = 0..50

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^(k*(k-1)/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 xrange(n)])

print map(a, xrange(21)) # Indranil Ghosh, Aug 12 2017

CROSSREFS

Sequence in context: A001908 A270703 A192547 * A244437 A265003 A193363

Adjacent sequences:  A006126 A006127 A006128 * A006130 A006131 A006132

KEYWORD

nonn,nice,easy

AUTHOR

Colin Mallows

EXTENSIONS

More terms from Vladeta Jovovic, Apr 09 2000

STATUS

approved

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Last modified September 19 14:21 EDT 2017. Contains 292242 sequences.