A129890 a(n) = (2*n+2)!! - (2*n+1)!!.

1, 5, 33, 279, 2895, 35685, 509985, 8294895, 151335135, 3061162125, 68000295825, 1645756410375, 43105900812975, 1214871076343925, 36659590336994625, 1179297174137457375, 40288002704636061375, 1456700757237661060125

Offset: 0

Author

Paolo P. Lava and Giorgio Balzarotti, Jun 04 2007

Keywords

• easy
• nonn

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

Examples

			2!! - 1!! =  2 -  1 =  1;
4!! - 3!! =  8 -  3 =  5;
6!! - 5!! = 48 - 15 = 33.
		

Links

• Selden Crary, Richard Diehl Martinez, and Michael Saunders, The Nu Class of Low-Degree-Truncated Rational Multifunctions. Ib. Integrals of Matern-correlation functions for all odd-half-integer class parameters, arXiv:1707.00705 [stat.ME], 2017, Table 2.
• Alexander Kreinin, Integer Sequences and Laplace Continued Fraction, Preprint 2016.
• Alexander Kreinin, Integer Sequences Connected to the Laplace Continued Fraction and Ramanujan's Identity, Journal of Integer Sequences, 19 (2016), #16.6.2.
• N. Ochiumi, On the total sum of number of nodes covering a given number of leaves in an unordered binary tree
• Donovan Young, A critical quartet for queuing couples, arXiv:2007.13868 [math.CO], 2020.

Crossrefs

Cf. A006882, A122649, A202212, A336599.

Programs

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 *)

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) = (2*n+1)*a(n-1) + A000165(n). - Philippe Deléham, Oct 28 2013

Conjecture: a(n) = (2*n + 2)*(2*n + 2)! * Sum_{k >= 1} (-1)^(k+1)/Product_{j = 0..n+1} (k + 2*j). - Peter Bala, Jul 06 2025

Extensions

New name from Peter Luschny, Dec 01 2014