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

%I A277586 #27 Feb 16 2025 08:33:37
%S A277586 1,4,22,32,488,5408,70544,23552,1202048,22846976,22850816,40431616,
%T A277586 2628156416,1576923136,228655904768,416962576384,2362792902656,
%U A277586 7088385949696,262270410489856,52454094798848,2150618140770304,92476585387491328,462382939977023488
%N A277586 Numerator of Sum_{k=0..n} (2^k * (k!)^2)/(2k + 1)!.
%C A277586 Let b(n) = Sum_{k=0..n} (2^k * (k!)^2)/(2k + 1)!. Then:
%C A277586 b(n) = 1 + 1/3 * (1 + 2/5 * (1 + … (1 + n/(2n+1)))) = A087547(n+1)/A001147(n+1).
%C A277586 lim n -> infinity b(n) = Pi/2.
%H A277586 Seiichi Manyama, <a href="/A277586/b277586.txt">Table of n, a(n) for n = 0..1154</a>
%H A277586 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PiFormulas.html">Pi Formulas</a>
%F A277586 a(n) = numerator(Sum_{k=0..n} (2^k)/A002457(k)).
%e A277586 b(0) = 1, so a(0) = 1.
%e A277586 b(1) = 4/3, so a(1) = 4.
%e A277586 b(2) = 22/15, so a(2) = 22.
%e A277586 b(3) = 32/21, so a(3) = 32.
%e A277586 b(4) = 488/315, so a(4) = 488.
%o A277586 (PARI) a(n) = numerator(sum(k=0, n, (2^k * (k!)^2)/(2*k + 1)!)); \\ _Michel Marcus_, Oct 22 2016
%Y A277586 Cf. A001147, A002457, A019669 (decimal expansion of Pi/2), A087547, A277585 (denominators).
%K A277586 nonn,frac
%O A277586 0,2
%A A277586 _Seiichi Manyama_, Oct 22 2016