A007946 a(n) = 6*(2*n+1)! / ((n!)^2*(n+3)).

2, 9, 36, 140, 540, 2079, 8008, 30888, 119340, 461890, 1790712, 6953544, 27041560, 105306075, 410605200, 1602881040, 6263890380, 24502865310, 95937144600, 375945078600, 1474358525640, 5786272150230, 22724268808176, 89301056353200, 351140573438200, 1381487341784004

Offset: 0

Author

David W. Wilson and Dean Hickerson, Apr 21 1997

Keywords

• nonn
• easy

Comments

If Y is a fixed 2-subset of a 2n-set X then a(n-2) is the number of (n-1)-subsets of X intersecting Y. - Milan Janjic, Oct 21 2007

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Milan Janjic, Two Enumerative Functions.

Crossrefs

Cf. A000120, A001622, A001791, A002054, A061549, A242986.

Programs

Magma

[Binomial(2*n+2, n) + Binomial(2*n+3, n) : n in [0..30]]; // Wesley Ivan Hurt, Aug 23 2014

Maple

A007946:=n->binomial(2*n+2,n)+binomial(2*n+3,n): seq(A007946(n), n=0..30); # Wesley Ivan Hurt, Aug 23 2014

Mathematica

Table[Binomial[2 n + 2, n] + Binomial[2 n + 3, n], {n, 0, 30}] (* Wesley Ivan Hurt, Aug 23 2014 *)
Table[6*(2*n + 1)!/((n!)^2*(n + 3)), {n,0,50}] (* G. C. Greubel, Jan 23 2017 *)

PARI

for(n=0,50, print1(6*(2*n + 1)!/((n!)^2*(n + 3)), ", ")) \\ G. C. Greubel, Jan 23 2017

Formula

a(n) = C(2n+2, n) + C(2n+3, n). - Emeric Deutsch, May 16 2003

From Karol A. Penson, Aug 23 2014: (Start)

O.g.f.: ((-2+1/z^2-2/z)/sqrt(1-4*z)-1/z^2)/(2*z).

Representation as the n-th moment of a signed function: w(x) = sqrt(x/(4-x))*(x^2-2*x-2)/(2*Pi) on the segment x = (0,4): a(n) = Integral_{x=0..4} x^n*w(x) dx. For x->0, w(x)->0, and for x->4, w(x)->infinity.

a(n) ~ (3/65536)*(4^n)*(-55332459+18443992*n - 6147840*n^2 + 2050048*n^3 - 688128*n^4 + 262144*n^5)/(n^(11/2)*sqrt(Pi)), for n->infinity.

(End)

a(n) = A001791(n+1) + A002054(n+1). - Wesley Ivan Hurt, Aug 23 2014

From Peter Luschny, Aug 25 2014: (Start)

a(n) = ((6*(2*n+1))/(n+3))* binomial(2*n,n).

a(n) has the asymptotic series 2^(2*n+3)*(1+(n+3)/((2*n+3))) *Sum_{k>=0}((num(k)/den(k))*(-n)^(-k))/sqrt(n*Pi). Here den(n) = 2^(4*n-A000120(n)) = A061549(n) and num(n) = 1, 25, 1297, 32755, 3249099, 79652055, 3876842453, 93900904955, 18138634602803, 437081823058595, 21036073578365391,... For example a(100) = 0.10602088220899083... *10^61 with the given values of num.

a(x) ~ exp(x*log(4)-(log(Pi)+cos(2*Pi*x)*(log(x) + 1/(4*x)))/2 + log((12*x+6)/ (3+x))). For example, this formula gives a(100) = 0.10602088... *10^61.

a(n) = A242986(2*n). (End)

a(n) = 12*4^n*Gamma(3/2+n)/(sqrt(Pi)*(3+n)*Gamma(1+n)). - Peter Luschny, Dec 14 2015

a(n) = 2*Sum_{i=0..n} (1/(i+1)*binomial(2*i+3,i+3)*binomial(2*(n-i),n-i)). - Vladimir Kruchinin, Apr 20 2016

E.g.f.: 2*(x*(-1 + 3*x)*BesselI(0,2*x) + (1 - 2*x + 3*x^2) * BesselI(1,2*x))*exp(2*x)/x^2. - Ilya Gutkovskiy, Apr 20 2016

D-finite with recurrence n*(n+3)*a(n) -2*(n+2)*(2*n+1)*a(n-1) = 0. - R. J. Mathar, Mar 30 2022

From Amiram Eldar, Feb 16 2023: (Start)

Sum_{n>=0} 1/a(n) = 8*Pi/(27*sqrt(3)) + 1/9.

Sum_{n>=0} (-1)^n/a(n) = 8*log(phi)/(5*sqrt(5)) + 1/15, where phi is the golden ratio (A001622). (End)