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

1, 8, 209, 10016, 822321, 98607816, 16772776929, 3755613340800, 1089481085841825, 392115220017568200, 173351482189397931825, 91513890536903699104800, 57296185618906061753900625, 41706416795344237885218165000, 35120660862575611007699136530625

Offset: 0

Author

Daniel Suteu, Jul 21 2016

Keywords

• easy
• nonn

Links

• Daniel Suteu, Table of n, a(n) for n = 0..99

Programs

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

Mathematica

Table[((2 n + 1)!!)^2 Sum[(-1)^k/(2 k + 1)^2, {k, 0, n}], {n, 0, 14}] (* Michael De Vlieger, Jul 21 2016 *)

PARI

dfo(n) = (2*n)! / n! / 2^n; \\ after A001147
a(n) = dfo(n+1)^2*sum(k=0, n, (-1)^k/(2*k+1)^2); \\ Michel Marcus, Jul 25 2016

Sidef

var k = 0
func a(n) { (-1)**n }
func b(n) { (2*n + 1)**2  }
func g((k))         { b(k) }
func g(n) is cached { b(n) * g(n-1) }
func f((k))         { a(k) }
func f(n) is cached { b(n)*f(n-1) + a(n)*g(n-1) }
for i in (k .. 20) { say f(i) }

Formula

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

a(n) ~ A006752 * ((2*n+1)!!)^2. - Daniel Suteu, Dec 03 2016