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A129890
a(n) = (2*n+2)!! - (2*n+1)!!.
12
1, 5, 33, 279, 2895, 35685, 509985, 8294895, 151335135, 3061162125, 68000295825, 1645756410375, 43105900812975, 1214871076343925, 36659590336994625, 1179297174137457375, 40288002704636061375, 1456700757237661060125
OFFSET
0,2
COMMENTS
Previous name was: Difference between the double factorial of the n-th nonnegative even number and the double factorial of the n-th nonnegative odd number.
In other words, a(n) = b(2n+2)-b(2n+1), where b = A006882. - N. J. A. Sloane, Dec 14 2011 [Corrected Peter Luschny, Dec 01 2014]
a(n) is the number of linear chord diagrams on 2n+2 vertices with one marked chord such that none of the remaining n chords are contained within the marked chord, see [Young]. - Donovan Young, Aug 11 2020
FORMULA
E.g.f.: 2/((1-2*x)^2)-1/[(1-2*x)*sqrt(1-2*x)]. - Sergei N. Gladkovskii, Dec 04 2011
a(n) = (2n+1)*a(n-1) + A000165(n). - Philippe Deléham, Oct 28 2013
EXAMPLE
2!! - 1!! = 2 - 1 = 1;
4!! - 3!! = 8 - 3 = 5;
6!! - 5!! = 48 - 15 = 33.
MAPLE
seq(doublefactorial(2*n+2)-doublefactorial(2*n+1), n=0..9); # Peter Luschny, Dec 01 2014
MATHEMATICA
a[n_] := (2n+2)!! - (2n+1)!!;
Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Jul 30 2018 *)
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
EXTENSIONS
New name from Peter Luschny, Dec 01 2014
STATUS
approved