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

%I A277585 #53 Feb 16 2025 08:33:37
%S A277585 1,3,15,21,315,3465,45045,15015,765765,14549535,14549535,25741485,
%T A277585 1673196525,1003917915,145568097675,265447707525,1504203675975,
%U A277585 4512611027925,166966608033225,33393321606645,1369126185872445,58872425992515135,294362129962575675
%N A277585 Denominator of Sum_{k=0..n} (2^k * (k!)^2)/(2k + 1)!.
%C A277585 Let b(n) = Sum_{k=0..n} (2^k * (k!)^2)/(2k + 1)!. Then:
%C A277585 b(n) = 1 + 1/3 * (1 + 2/5 * (1 + … (1 + n/(2n+1)))) = A087547(n+1)/A001147(n+1).
%C A277585 lim n -> infinity b(n) = Pi/2.
%H A277585 Seiichi Manyama, <a href="/A277585/b277585.txt">Table of n, a(n) for n = 0..1154</a>
%H A277585 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PiFormulas.html">Pi Formulas</a>
%F A277585 a(n) = denominator(Sum_{k=0..n} (2^k)/A002457(k)).
%e A277585 b(0) = 1, so a(0) = 1.
%e A277585 b(1) = 4/3, so a(1) = 3.
%e A277585 b(2) = 22/15, so a(2) = 15.
%e A277585 b(3) = 32/21, so a(3) = 21.
%e A277585 b(4) = 488/315, so a(4) = 315.
%o A277585 (PARI) a(n) = denominator(sum(k=0, n, (2^k * (k!)^2)/(2*k + 1)!)); \\ _Michel Marcus_, Oct 22 2016
%Y A277585 Cf. A001147, A002457, A019669 (decimal expansion of Pi/2), A087547, A277586 (numerators).
%K A277585 nonn,frac
%O A277585 0,2
%A A277585 _Seiichi Manyama_, Oct 22 2016