OFFSET
0,2
COMMENTS
Conjecture: Also the number of loop-graphs on n vertices without any non-loop edge having loops at both ends, with formula a(n) = Sum_{k=0..n} binomial(n,k) 2^(k*(n-k) + binomial(k,2)). The unlabeled version is A339832. - Gus Wiseman, Jan 25 2024
The above conjecture is true since (n-k)*k + binomial(n-k,2) = binomial(n,2) - binomial(k,2) and A006125 gives the denominators for this sequence. - Andrew Howroyd, Feb 20 2024
REFERENCES
D. L. Kreher and D. R. Stinson, Combinatorial Algorithms, CRC Press, 1999, p. 113.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..80
FORMULA
E.g.f.: Sum_{n>=0} a(n)*x^n/n! = Sum_{n>=0} exp(2^n*x)*2^(n(n-1)/2)*x^n/n!. - Paul D. Hanna, Sep 14 2009
a(n) = Sum_{k=0..n} binomial(n,k) * 2^(binomial(n,2)-binomial(k,2)). - Andrew Howroyd, Feb 20 2024
EXAMPLE
1, 2, 7/2, 45/8, 545/64, 12625/1024, 564929/32768, 49162689/2097152, ...
MAPLE
f := n->add(binomial(n, k)/2^(k*(k-1)/2), k=0..n);
MATHEMATICA
Table[Numerator[Sum[Binomial[n, k]/2^Binomial[k, 2], {k, 0, n}]], {n, 0, 20}] (* G. C. Greubel, Jun 19 2019 *)
PROG
(PARI) {a(n)=n!*polcoeff(sum(k=0, n, exp(2^k*x +x*O(x^n))*2^(k*(k-1)/2)*x^k/k!), n)} \\ Paul D. Hanna, Sep 14 2009
(PARI) a(n) = sum(k=0, n, binomial(n, k)*2^(binomial(n, 2)-binomial(k, 2))) \\ Andrew Howroyd, Feb 20 2024
(Magma) [Numerator( (&+[Binomial(n, k)/2^Binomial(k, 2): k in [0..n]]) ): n in [0..20]]; // G. C. Greubel, Jun 19 2019
(Sage) [numerator( sum(binomial(n, k)/2^binomial(k, 2) for k in (0..n)) ) for n in (0..20)] # G. C. Greubel, Jun 19 2019
CROSSREFS
KEYWORD
nonn,frac
AUTHOR
N. J. A. Sloane, Jan 20 2003
STATUS
approved