A277586 Numerator of Sum_{k=0..n} (2^k * (k!)^2)/(2k + 1)!.

1, 4, 22, 32, 488, 5408, 70544, 23552, 1202048, 22846976, 22850816, 40431616, 2628156416, 1576923136, 228655904768, 416962576384, 2362792902656, 7088385949696, 262270410489856, 52454094798848, 2150618140770304, 92476585387491328, 462382939977023488

Offset

0,2

Comments

Let b(n) = Sum_{k=0..n} (2^k * (k!)^2)/(2k + 1)!. Then:

b(n) = 1 + 1/3 * (1 + 2/5 * (1 + … (1 + n/(2n+1)))) = A087547(n+1)/A001147(n+1).

lim n -> infinity b(n) = Pi/2.

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1154

Eric Weisstein's World of Mathematics, Pi Formulas

Formula

a(n) = numerator(Sum_{k=0..n} (2^k)/A002457(k)).

Example

b(0) = 1, so a(0) = 1.
b(1) = 4/3, so a(1) = 4.
b(2) = 22/15, so a(2) = 22.
b(3) = 32/21, so a(3) = 32.
b(4) = 488/315, so a(4) = 488.

Prog

(PARI) a(n) = numerator(sum(k=0, n, (2^k * (k!)^2)/(2*k + 1)!)); \\ Michel Marcus, Oct 22 2016

Crossrefs

Cf. A001147, A002457, A019669 (decimal expansion of Pi/2), A087547, A277585 (denominators).

Sequence in context: A036920 A036921 A083510 * A078647 A031108 A350521

Adjacent sequences: A277583 A277584 A277585 * A277587 A277588 A277589

Keyword

nonn,frac

Author

Seiichi Manyama, Oct 22 2016

Status

approved