|
| |
|
|
A130905
|
|
E.g.f.: exp(x^2 / 2) / (1 - x).
|
|
6
| |
|
|
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; 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. [From Austin Shapiro (auspex(AT)umich.edu), 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
|
|
|
FORMULA
| E.g.f.: exp(x^2 / 2) / (1 - x) = exp( x^2 / 2 + Sum_{k>0} x^k / k ).
|
|
|
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]
|
|
|
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: A180741 A121101 A080635 * A030799 A058105 A090012
Adjacent sequences: A130902 A130903 A130904 * A130906 A130907 A130908
|
|
|
KEYWORD
| nonn
|
|
|
AUTHOR
| Karol A. Penson (penson(AT)lptl.jussieu.fr), Jun 08 2007
|
|
|
EXTENSIONS
| Superfluous leading 1 deleted by Johannes W. Meijer, Jul 21 2011
|
| |
|
|
