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A275286 a(n) = ((2n+1)!!)^2 * Sum_{k=0..n}(-1)^k/(2k+1)^2. 2
1, 8, 209, 10016, 822321, 98607816, 16772776929, 3755613340800, 1089481085841825, 392115220017568200, 173351482189397931825, 91513890536903699104800, 57296185618906061753900625, 41706416795344237885218165000, 35120660862575611007699136530625 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

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

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

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 *)

PROG

(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) }

(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

(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

CROSSREFS

Sequence in context: A330287 A279663 A294970 * A255854 A151797 A279464

Adjacent sequences:  A275283 A275284 A275285 * A275287 A275288 A275289

KEYWORD

easy,nonn

AUTHOR

Daniel Suteu, Jul 21 2016

STATUS

approved

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Last modified January 16 05:22 EST 2022. Contains 350374 sequences. (Running on oeis4.)