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%I #37 Jan 16 2024 01:38:17
%S 1,4,48,640,8960,129024,1892352,28114944,421724160,6372720640,
%T 96865353728,1479398129664,22684104654848,348986225459200,
%U 5384358907084800,83278084429578240,1290810308658462720,20045524793284362240,311819274562201190400,4857816066863765913600
%N a(n) = (0^n + 4^n * binomial(2*n,n))/2.
%C 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
%C Tested up to and including a(210). - _R. H. Hardin_, May 08 2013
%H G. C. Greubel, <a href="/A098402/b098402.txt">Table of n, a(n) for n = 0..825</a>
%F G.f.: 8*x/( sqrt(1 - 16*x)*(1 - sqrt(1 - 16*x)) ).
%F a(n+1) = 4*A098400(n).
%F n*a(n) - 8*(2*n-1)*a(n-1) = 0. - _R. J. Mathar_, Nov 09 2012
%F a(n) ~ 16^n/(2*sqrt(Pi*n)). - _Ilya Gutkovskiy_, Aug 03 2016
%F a(n) = A055372(2*n,n). - _Alois P. Heinz_, Jan 21 2020
%F From _Amiram Eldar_, Jan 16 2024: (Start)
%F Sum_{n>=0} 1/a(n) = 17/15 + 32*arcsin(1/4)/(15*sqrt(15)).
%F Sum_{n>=0} (-1)^n/a(n) = 15/17 - 32*arcsinh(1/4)/(17*sqrt(17)). (End)
%t Table[(Boole[n == 0] + 4^n Binomial[2 n, n])/2, {n, 0, 18}] (* or *)
%t CoefficientList[Series[8 x/(# (1 - #)) &@ Sqrt[1 - 16 x], {x, 0, 18}], x] (* _Michael De Vlieger_, Aug 03 2016 *)
%o (Magma) [(0^n + 4^n*(n+1)*Catalan(n))/2: n in [0..40]]; // _G. C. Greubel_, Dec 27 2023
%o (SageMath) [(4^n*binomial(2*n,n) + int(n==0))/2 for n in range(41)] # _G. C. Greubel_, Dec 27 2023
%Y Cf. A055372, A069723, A069720, A098400, A098401.
%K easy,nonn
%O 0,2
%A _Paul Barry_, Sep 06 2004
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