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A275286 a(n) = ((2n+1)!!)^2 * Sum_{k=0..n}(-1)^k/(2k+1)^2.

%I A275286 #28 Sep 08 2022 08:46:17
%S A275286 1,8,209,10016,822321,98607816,16772776929,3755613340800,
%T A275286 1089481085841825,392115220017568200,173351482189397931825,
%U A275286 91513890536903699104800,57296185618906061753900625,41706416795344237885218165000,35120660862575611007699136530625
%N A275286 a(n) = ((2n+1)!!)^2 * Sum_{k=0..n}(-1)^k/(2k+1)^2.
%H A275286 Daniel Suteu, <a href="/A275286/b275286.txt">Table of n, a(n) for n = 0..99</a>
%F A275286 a(0) = 1, a(n) = (2n+1)^2 * a(n-1) + (-1)^n / 4^n * ((2n+1)!)^2 / (n!)^2 / (2n+1)^2. - _Daniel Suteu_, Jul 21 2016
%F A275286 a(n) ~ A006752 * ((2*n+1)!!)^2. - _Daniel Suteu_, Dec 03 2016
%t A275286 Table[((2 n + 1)!!)^2 Sum[(-1)^k/(2 k + 1)^2, {k, 0, n}], {n, 0, 14}] (* _Michael De Vlieger_, Jul 21 2016 *)
%o A275286 (Sidef)
%o A275286 var k = 0
%o A275286 func a(n) { (-1)**n }
%o A275286 func b(n) { (2*n + 1)**2  }
%o A275286 func g((k))         { b(k) }
%o A275286 func g(n) is cached { b(n) * g(n-1) }
%o A275286 func f((k))         { a(k) }
%o A275286 func f(n) is cached { b(n)*f(n-1) + a(n)*g(n-1) }
%o A275286 for i in (k .. 20) { say f(i) }
%o A275286 (PARI) dfo(n) = (2*n)! / n! / 2^n; \\ after A001147
%o A275286 a(n) = dfo(n+1)^2*sum(k=0, n, (-1)^k/(2*k+1)^2); \\ _Michel Marcus_, Jul 25 2016
%o A275286 (Magma) [(Factorial(2*n+1)/(2^n*Factorial(n)))^2*(&+[(-1)^k/(2*k+1)^2: k in [0..n]]): n in [0..20]]; // _G. C. Greubel_, Aug 25 2018
%K A275286 easy,nonn
%O A275286 0,2
%A A275286 _Daniel Suteu_, Jul 21 2016