This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A079491 #23 Feb 20 2024 18:37:29
%S A079491 1,2,7,45,545,12625,564929,49162689,8361575425,2789624383745,
%T A079491 1830776926245889,2368773751202917377,6053217182280501452801,
%U A079491 30595465072175429929979905,306239118989330960523869667329,6076268165073202122463201684865025
%N A079491 Numerator of Sum_{k=0..n} binomial(n,k)/2^(k*(k-1)/2).
%C A079491 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
%C A079491 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
%D A079491 D. L. Kreher and D. R. Stinson, Combinatorial Algorithms, CRC Press, 1999, p. 113.
%H A079491 G. C. Greubel, <a href="/A079491/b079491.txt">Table of n, a(n) for n = 0..80</a>
%F A079491 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
%F A079491 a(n) = Sum_{k=0..n} binomial(n,k) * 2^(binomial(n,2)-binomial(k,2)). - _Andrew Howroyd_, Feb 20 2024
%e A079491 1, 2, 7/2, 45/8, 545/64, 12625/1024, 564929/32768, 49162689/2097152, ...
%p A079491 f := n->add(binomial(n,k)/2^(k*(k-1)/2),k=0..n);
%t A079491 Table[Numerator[Sum[Binomial[n,k]/2^Binomial[k,2], {k,0,n}]], {n,0,20}] (* _G. C. Greubel_, Jun 19 2019 *)
%o A079491 (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
%o A079491 (PARI) a(n) = sum(k=0, n, binomial(n,k)*2^(binomial(n,2)-binomial(k,2))) \\ _Andrew Howroyd_, Feb 20 2024
%o A079491 (Magma) [Numerator( (&+[Binomial(n,k)/2^Binomial(k,2): k in [0..n]]) ): n in [0..20]]; // _G. C. Greubel_, Jun 19 2019
%o A079491 (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
%Y A079491 Denominators are in A006125.
%Y A079491 Cf. A079492.
%Y A079491 The unlabeled version is A339832 (loop-graphs interpretation).
%Y A079491 A000085, A100861, A111924 count set partitions into singletons or pairs.
%Y A079491 A000666 counts unlabeled loop-graphs, covering A322700.
%Y A079491 A006125 (shifted left) counts labeled loop-graphs, covering A322661.
%Y A079491 A006129 counts labeled covering graphs, connected A001187.
%K A079491 nonn,frac
%O A079491 0,2
%A A079491 _N. J. A. Sloane_, Jan 20 2003