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A130905 E.g.f.: exp(x^2 / 2) / (1 - x). 13
1, 1, 3, 9, 39, 195, 1185, 8295, 66465, 598185, 5982795, 65810745, 789739335, 10266611355, 143732694105, 2155990411575, 34495848612225, 586429426407825, 10555729709800275, 200558864486205225, 4011177290378833575 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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

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.

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

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

EXAMPLE

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.

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 *)

PROG

(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 */

CROSSREFS

Sequence in context: A080635 A278749 A208816 * A030799 A273396 A058105

Adjacent sequences:  A130902 A130903 A130904 * A130906 A130907 A130908

KEYWORD

nonn

AUTHOR

Karol A. Penson, Jun 08 2007

EXTENSIONS

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

STATUS

approved

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Last modified October 16 02:45 EDT 2018. Contains 316252 sequences. (Running on oeis4.)