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A006129
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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)
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154
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1, 0, 1, 4, 41, 768, 27449, 1887284, 252522481, 66376424160, 34509011894545, 35645504882731588, 73356937912127722841, 301275024444053951967648, 2471655539737552842139838345, 40527712706903544101000417059892, 1328579255614092968399503598175745633
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OFFSET
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0,4
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COMMENTS
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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
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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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
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EXAMPLE
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2^binomial(n,2) = 1 + binomial(n,2) + 4*binomial(n,3) + 41*binomial(n,4) + 768*binomial(n,5) + ...
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MAPLE
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a:= proc(n) option remember; `if`(n=0, 1,
2^binomial(n, 2) - add(a(k)*binomial(n, k), k=0..n-1))
end:
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MATHEMATICA
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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 *)
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PROG
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(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))
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CROSSREFS
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KEYWORD
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nonn,nice,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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