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 A130905 #44 Feb 27 2024 09:47:35
%S A130905 1,1,3,9,39,195,1185,8295,66465,598185,5982795,65810745,789739335,
%T A130905 10266611355,143732694105,2155990411575,34495848612225,
%U A130905 586429426407825,10555729709800275,200558864486205225,4011177290378833575
%N A130905 Expansion of e.g.f. exp(x^2 / 2) / (1 - x).
%C A130905 a(n) is also the number of oriented simple graphs on n labeled vertices, such that each weakly connected component with 3 or more vertices is a directed cycle. - _Austin Shapiro_, Apr 17 2009
%C A130905 The Kn2p sums, p>=1, see A180662 for the definition of these sums, of triangle A193229 lead to this sequence. - _Johannes W. Meijer_, Jul 21 2011
%C A130905 Compare with A000266 with e.g.f. exp( -x^2 / 2) / (1 - x). - _Michael Somos_, Jul 24 2011
%C A130905 a(n) is the number of permutations of an n-set where each transposition (two cycle) is counted twice. That is, each transposition is an involution and is its own inverse, but if we imagine each transposition can be oriented in one of two ways, then a permutation with oriented transpositions is just a oriented simple graph. Conversely, an oriented simple graph with restrictions on connected components comes from a permutation with oriented transpositions. - _Michael Somos_, Jul 25 2011
%H A130905 Vincenzo Librandi, <a href="/A130905/b130905.txt">Table of n, a(n) for n = 0..200</a>
%H A130905 Michael Wallner, <a href="https://arxiv.org/abs/1706.07163">A bijection of plane increasing trees with relaxed binary trees of right height at most one</a>, arXiv:1706.07163 [math.CO], 2017, Table 2 on p. 13.
%F A130905 E.g.f.: exp(x^2/2) / (1-x) = exp( x^2 / 2 + sum(k>=1, x^k/k ) ).
%F A130905 E.g.f.: 1/E(0) where E(k)=1 - x/(1 - x/(x + (2*k+2)/E(k+1))); (continued fraction, 3rd kind, 3-step). - _Sergei N. Gladkovskii_, Sep 20 2012
%F A130905 D-finite with recurrence: a(n) = n*a(n-1) + (n-1)*a(n-2) - (n-2)*(n-1)*a(n-3) . - _Vaclav Kotesovec_, Oct 20 2012
%F A130905 a(n) ~ n!*exp(1/2) . - _Vaclav Kotesovec_, Oct 20 2012
%F A130905 E.g.f.: E(0)/(1-x)^2, where E(k)= 1 - x/(1 - x/(x - 2*(k+1)/E(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Jul 10 2013
%F A130905 a(n) = n! * Sum_{k=0..floor(n/2)} 1/(2^k * k!). - _Seiichi Manyama_, Feb 27 2024
%e A130905 1 + x + 3*x^2 + 9*x^3 + 39*x^4 + 195*x^5 + 1185*x^6 + 8295*x^7 + ...
%e A130905 a(2) = 3 because there are 3 oriented simple graphs on two labeled vertices. a(3) = 9 because for oriented simple graphs on three labeled vertices there is 1 with no edges, 6 with one edge, 0 with two edges, and 2 with three edges which are directed cycles such that each weakly connected component with 3 or more vertices is a directed cycle.
%p A130905 A130905 := proc(n) local x: n!*coeftayl(exp(x^2/2)/(1-x), x=0, n) end: seq(A130905(n), n=0..25); # _Johannes W. Meijer_, Jul 21 2011
%t A130905 CoefficientList[Series[E^(x^2/2)/(1-x), {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Oct 20 2012 *)
%o A130905 (PARI) {a(n) = if( n<0, 0, n! * polcoeff( exp( x^2 / 2 + x * O(x^n)) / (1 - x), n))} /* _Michael Somos_, Jul 24 2011 */
%Y A130905 Cf. A000522, A130906.
%Y A130905 Cf. A000266, A355268.
%K A130905 nonn
%O A130905 0,3
%A A130905 _Karol A. Penson_, Jun 08 2007
%E A130905 Superfluous leading 1 deleted by _Johannes W. Meijer_, Jul 21 2011