A098402 a(n) = (0^n + 4^n * binomial(2*n,n))/2.

1, 4, 48, 640, 8960, 129024, 1892352, 28114944, 421724160, 6372720640, 96865353728, 1479398129664, 22684104654848, 348986225459200, 5384358907084800, 83278084429578240, 1290810308658462720, 20045524793284362240, 311819274562201190400, 4857816066863765913600

Offset: 0

Author

Paul Barry, Sep 06 2004

Keywords

• easy
• nonn

Comments

It seems that a(n) is the number of pairs of binary vectors of length 2*n which are orthogonal. (Define binary vectors here to have components of value +1 or -1. There are no pairs of binary vectors of odd length which are orthogonal.) For example, the a(1) = 4 pairs of binary vectors of length 2 are (-1,-1) and (1,-1), (-1,-1) and (-1,1), (1,-1) and (1,1), (-1,1) and (1,1). Tested up to and including a(8). - R. J. Mathar, Apr 15 2013

Tested up to and including a(210). - R. H. Hardin, May 08 2013

Links

• G. C. Greubel, Table of n, a(n) for n = 0..825

Crossrefs

Cf. A055372, A069723, A069720, A098400, A098401.

Programs

Magma

[(0^n + 4^n*(n+1)*Catalan(n))/2: n in [0..40]]; // G. C. Greubel, Dec 27 2023

Mathematica

Table[(Boole[n == 0] + 4^n Binomial[2 n, n])/2, {n, 0, 18}] (* or *)
CoefficientList[Series[8 x/(# (1 - #)) &@ Sqrt[1 - 16 x], {x, 0, 18}], x] (* Michael De Vlieger, Aug 03 2016 *)

SageMath

[(4^n*binomial(2*n,n) + int(n==0))/2 for n in range(41)] # G. C. Greubel, Dec 27 2023

Formula

G.f.: 8*x/( sqrt(1 - 16*x)*(1 - sqrt(1 - 16*x)) ).

a(n+1) = 4*A098400(n).

n*a(n) - 8*(2*n-1)*a(n-1) = 0. - R. J. Mathar, Nov 09 2012

a(n) ~ 16^n/(2*sqrt(Pi*n)). - Ilya Gutkovskiy, Aug 03 2016

a(n) = A055372(2*n,n). - Alois P. Heinz, Jan 21 2020

From Amiram Eldar, Jan 16 2024: (Start)

Sum_{n>=0} 1/a(n) = 17/15 + 32*arcsin(1/4)/(15*sqrt(15)).

Sum_{n>=0} (-1)^n/a(n) = 15/17 - 32*arcsinh(1/4)/(17*sqrt(17)). (End)