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%I #24 Jun 01 2024 18:02:56
%S 2,32,864,36864,2304000,199065600,22759833600,3329438515200,
%T 606790169395200,134842259865600000,35895009576222720000,
%U 11277559372311429120000,4129466323494701629440000,1743270091026070964797440000,840505222458998500884480000000
%N a(n) = n * 2^n * (n!)^2.
%C Sum_{n>=1} a(n) / (2*n)! = Pi + 3.
%H Simon Plouffe, <a href="https://arxiv.org/abs/0912.0303">On the computation of the n'th decimal digit of various transcendental numbers</a>, arXiv:0912.0303 [math.NT], 2009.
%e a(2) = 2 * 2^2 * ( 2! )^2 = 2 * 4 * 4 = 32.
%e a(3) = 3 * 2^3 * ( 3! )^2 = 3 * 8 * 36 = 864.
%e Sum_{n=1..10} a(n) / ( 2n )! = 3 + 3.01310...
%e Sum_{n=1..12} a(n) / ( 2n )! = 3 + 3.10046...
%e Sum_{n=1..18} a(n) / ( 2n )! = 3 + 3.14046...
%e Sum_{n=1..20} a(n) / ( 2n )! = 3 + 3.14126...
%e Sum_{n=1..23} a(n) / ( 2n )! = 3 + 3.14154...
%t Table[n*2^n*(n!)^2,{n,20}] (* _Harvey P. Dale_, Jun 01 2024 *)
%o (Rexx)
%o S = 2
%o do N = 2 while length( S ) < 255
%o S = S || ', ' || N * ( 2 ** N ) * ( !( N ) ** 2 )
%o end N
%o say S ; return S
%Y Cf. A001044 ( (n!)^2 ), A010050 ( (2n)! ), A000796 (digits of Pi).
%K nonn
%O 1,1
%A _Frank Ellermann_, Apr 05 2020