login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A007946 a(n) = 6*(2*n+1)! / ((n!)^2*(n+3)). 9
2, 9, 36, 140, 540, 2079, 8008, 30888, 119340, 461890, 1790712, 6953544, 27041560, 105306075, 410605200, 1602881040, 6263890380, 24502865310, 95937144600, 375945078600, 1474358525640, 5786272150230, 22724268808176 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

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

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)/Pi on the segment x=(0,4). In Maple notation: a(n) = int(x^n*w(x), x=0..4). 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

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 *)

PROG

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

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

CROSSREFS

Cf. A001791, A002054.

Sequence in context: A073989 A288834 A134759 * A135593 A027995 A077836

Adjacent sequences:  A007943 A007944 A007945 * A007947 A007948 A007949

KEYWORD

nonn

AUTHOR

David W. Wilson and Dean Hickerson, Apr 21 1997

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified November 18 22:39 EST 2019. Contains 329305 sequences. (Running on oeis4.)