A277585 Denominator of Sum_{k=0..n} (2^k * (k!)^2)/(2k + 1)!.

1, 3, 15, 21, 315, 3465, 45045, 15015, 765765, 14549535, 14549535, 25741485, 1673196525, 1003917915, 145568097675, 265447707525, 1504203675975, 4512611027925, 166966608033225, 33393321606645, 1369126185872445, 58872425992515135, 294362129962575675

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

Example

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

Prog

(PARI) a(n) = denominator(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, A277586 (numerators).

Sequence in context: A230658 A117801 A347303 * A318134 A087674 A212846

Adjacent sequences: A277582 A277583 A277584 * A277586 A277587 A277588

Keyword

nonn,frac

Author

Seiichi Manyama, Oct 22 2016

Status

approved