A122649 -id:A122649 - cp's OEIS frontend

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

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

Offset: 0

Paolo P. Lava and Giorgio Balzarotti, Jun 04 2007

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

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

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.

Cf. A006882, A122649, A202212, A336599.

seq(doublefactorial(2*n+2)-doublefactorial(2*n+1),n=0..9); # Peter Luschny, Dec 01 2014

a[n_] := (2n+2)!! - (2n+1)!!;
Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Jul 30 2018 *)

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

New name from Peter Luschny, Dec 01 2014

A232618 a(n) = (2n)!! mod (2n-1)!! where k!! = A006882(k).

Original entry on oeis.org

0, 2, 3, 69, 60, 4500, 104580, 186795, 13497435, 442245825, 13003053525, 64585694250, 3576632909850, 147580842959550, 5708173568847750, 27904470362393625, 2292043480058957625, 126842184377462428875, 6371504674680470700375, 312265748715684068930625

Offset: 1

Alex Ratushnyak, Nov 27 2013

nonn

(2n)!! is the product of first n even numbers, (2n-1)!! is the product of first n odd numbers.

			a(3) = A006882(6) mod A006882(5) = 2*4*6 mod 1*3*5 = 48 mod 15 = 3.

Cf. A006882, A122649, A129890, A232617.

Table[Mod[(2n)!!,(2n-1)!!],{n,20}] (* Harvey P. Dale, Sep 23 2020 *)

o=e=1
for n in range(1,99,2):
  o*=n
  e*=n+1
  print(str(e%o), end=',')

a(n) = A006882(2*n) mod A006882(2*n-1).

A232617 Product of first n odd numbers plus product of first n even numbers: (2n-1)!! + (2n)!!, where k!! = A006882(k).

Original entry on oeis.org

3, 11, 63, 489, 4785, 56475, 780255, 12348945, 220253985, 4370620275, 95498916975, 2278224696825, 58917607974225, 1641787169697675, 49040157044253375, 1563094742062478625, 52953322446161762625, 1899986948191060603875, 71977860935783603175375, 2870913642898706235455625

Offset: 1

Alex Ratushnyak, Nov 27 2013

nonn

			a(3) = 1*3*5 + 2*4*6 = 15 + 48 = 63.

Cf. A006882, A122649, A129890.

Table[n!!+(n+1)!!,{n,1,41,2}] (* Harvey P. Dale, Jan 22 2019 *)

a(n)=prod(i=1,n,2*i-1)+prod(i=1,n,2*i) \\ Ralf Stephan, Nov 28 2013

o=e=1
for n in range(1,99,2):
  o*=n
  e*=n+1
  print(str(e+o), end=',')

a(n) = A006882(2*n-1) + A006882(2*n).
a(n) = A001147(n) + A000165(n).
a(n) +(-4*n+3)*a(n-1) +2*(n-1)*(2*n-3)*a(n-2)=0. - R. J. Mathar, Nov 23 2014

A306184 a(n) = (2n+1)!! mod (2n)!! where k!! = A006882(k).

Original entry on oeis.org

1, 7, 9, 177, 2715, 42975, 91665, 3493665, 97345395, 2601636975, 70985324025, 57891366225, 9411029102475, 476966861546175, 20499289200014625, 847876038362978625, 35160445175104123875, 1487419121780448231375, 945654757149212735625, 357657177058846280240625

Offset: 1

Alex Ratushnyak, Jan 27 2019

nonn

a(n) is divisible by A049606(n). - Robert Israel, Jan 28 2019

			a(3) = A006882(7) mod A006882(6) = (7*5*3) mod (6*4*2) = 105 mod 48 = 9.

Robert Israel, Table of n, a(n) for n = 1..403

Cf. A006882, A049606, A122649, A129890, A232617, A232618, A306185.

f:= n -> doublefactorial(2*n+1) mod doublefactorial(2*n):
map(f, [$1..40]); # Robert Israel, Jan 28 2019

Mod[#[[2]],#[[1]]]&/@Partition[Range[2,42]!!,2] (* Harvey P. Dale, May 29 2025 *)

o=e=1
for n in range(2, 99, 2):
  o*=n+1
  e*=n
  print(o%e, end=', ')

a(n) = A006882(2*n+1) mod A006882(2*n).

A306185 a(n) = (2n+1)!! + (2n)!! where k!! = A006882(k).

Original entry on oeis.org

5, 23, 153, 1329, 14235, 181215, 2672145, 44781345, 840523635, 17465201775, 397983749625, 9867844134225, 264469801070475, 7618612476650175, 234748657653134625, 7703855828862818625, 268263758052098683875, 9879138385352252391375, 383608053176023482431625, 15664153113813817068080625

Offset: 1

Alex Ratushnyak, Jan 27 2019

nonn

			a(3) = A006882(7) + A006882(6) = (7*5*3) + (6*4*2) = 105 + 48 = 153.

Cf. A006882, A122649, A129890, A232617, A232618, A306184.

o=e=1
for n in range(2, 99, 2):
  o*=n+1
  e*=n
  print(o+e, end=', ')

a(n) = A006882(2*n+1) + A006882(2*n).

cp's OEIS Frontend

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

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A232618 a(n) = (2n)!! mod (2n-1)!! where k!! = A006882(k).

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A232617 Product of first n odd numbers plus product of first n even numbers: (2n-1)!! + (2n)!!, where k!! = A006882(k).

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A306184 a(n) = (2n+1)!! mod (2n)!! where k!! = A006882(k).

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Maple

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A306185 a(n) = (2n+1)!! + (2n)!! where k!! = A006882(k).

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Examples

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Programs

Python

Formula

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