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A006129 a(0),a(1),a(2),... satisfy Sum a(k) binomial(n,k) (k=0..n) = 2^binomial(n,2), for n=0,1,...
(Formerly M3678)
15
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).

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

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]

Table[Sum[(-1)^(n-k) * Binomial[n, k] * 2^(k*(k-1)/2), {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, May 04 2015 *)

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 and additional comments from Vladeta Jovovic, Apr 09 2000

STATUS

approved

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Last modified December 9 23:16 EST 2016. Contains 278993 sequences.