A130905 Expansion of e.g.f. exp(x^2 / 2) / (1 - x).

1, 1, 3, 9, 39, 195, 1185, 8295, 66465, 598185, 5982795, 65810745, 789739335, 10266611355, 143732694105, 2155990411575, 34495848612225, 586429426407825, 10555729709800275, 200558864486205225, 4011177290378833575

Offset: 0

Author

Karol A. Penson, Jun 08 2007

Keywords

• nonn

Comments

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

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

Compare with A000266 with e.g.f. exp( -x^2 / 2) / (1 - x). - Michael Somos, Jul 24 2011

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

Examples

			1 + x + 3*x^2 + 9*x^3 + 39*x^4 + 195*x^5 + 1185*x^6 + 8295*x^7 + ...
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.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Michael Wallner, A bijection of plane increasing trees with relaxed binary trees of right height at most one, arXiv:1706.07163 [math.CO], 2017, Table 2 on p. 13.

Crossrefs

Cf. A000522, A130906.

Cf. A000266, A355268.

Programs

Maple

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

Mathematica

CoefficientList[Series[E^(x^2/2)/(1-x), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 20 2012 *)

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 */

Formula

E.g.f.: exp(x^2/2) / (1-x) = exp( x^2 / 2 + sum(k>=1, x^k/k ) ).

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

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

a(n) ~ n!*exp(1/2) . - Vaclav Kotesovec, Oct 20 2012

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

a(n) = n! * Sum_{k=0..floor(n/2)} 1/(2^k * k!). - Seiichi Manyama, Feb 27 2024

Extensions

Superfluous leading 1 deleted by Johannes W. Meijer, Jul 21 2011