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A053987 Numerators of successive convergents to tan(1/2) using continued fraction 1/(2-1/(6-1/(10-1/(14-1/(18-1/(22-1/(26-1/30-...))))))).

%I A053987 #58 Sep 08 2022 08:45:00
%S A053987 1,6,59,820,14701,322602,8372951,250865928,8521068601,323549740910,
%T A053987 13580568049619,624382580541564,31205548459028581,1684475234207001810,
%U A053987 97668358035547076399,6053753722969711734928,399450077357965427428849,27955451661334610208284502
%N A053987 Numerators of successive convergents to tan(1/2) using continued fraction 1/(2-1/(6-1/(10-1/(14-1/(18-1/(22-1/(26-1/30-...))))))).
%H A053987 Vincenzo Librandi, <a href="/A053987/b053987.txt">Table of n, a(n) for n = 1..200</a>
%H A053987 S. Janson, <a href="https://hal.inria.fr/hal-00990430">A divergent generating function that can be summed and analysed analytically</a>, Discrete Mathematics and Theoretical Computer Science; 2010, Vol. 12, No. 2, 1-22.
%F A053987 a(n) = Sum_{k=0..floor((n-1)/2)} (-1)^k*(2*n-2*k-1)!/((n-2*k-1)! * (2*k+1)!). - _Benoit Cloitre_, Jan 03 2006
%F A053987 E.g.f.: 1-cos(x*C(x)), C(x)=(1-sqrt(1-4*x))/(2*x) (A000108). - _Vladimir Kruchinin_, Aug 10 2010
%F A053987 From _Peter Bala_, Aug 01 2013, (Start)
%F A053987 a(n+1) = (4*n+2)*a(n) - a(n-1) with a(0) = 0 and a(1) = 1.
%F A053987 a(n) = Sum_{k = 0..floor((n-1)/2)} (-1)^k*4^(n-2*k-1)*(n-2*k-1)!*binomial(n-k-1, k)*binomial(n-k-1/2, k+1/2), see A058798. (End)
%F A053987 a(n) ~ sin(1/2) * 2^(2*n+1/2) * n^n / exp(n). - _Vaclav Kotesovec_, Feb 25 2014
%F A053987 a(n) = 4^n*Gamma(n+1/2)*hypergeometric([1/2-n/2,1-n/2], [3/2,1/2-n,1-n], -1/4)/sqrt(4*Pi). - _Peter Luschny_, Sep 10 2014
%p A053987 A053987 := n -> local k; add((-1)^k*(2*n-2*k-1)!/((n-2*k-1)!*(2*k+1)!), k = 0..floor((n-1)/2)); seq(A053987(n), n = 1..20); # _G. C. Greubel_, May 17 2020
%t A053987 Rest[CoefficientList[Series[Sin[(1-Sqrt[1-4*x])/2]/Sqrt[1-4*x], {x, 0, 20}], x] * Range[0, 20]!] (* _Vaclav Kotesovec_, Feb 25 2014 *)
%o A053987 (PARI) a(n)=sum(k=0,floor((n-1)/2),(-1)^k*(2*n-2*k-1)!/(n-2*k-1)!/(2*k+1)!) \\ _Benoit Cloitre_, Jan 03 2006
%o A053987 (Sage)
%o A053987 def A053987(n):
%o A053987     return 4^n*gamma(n+1/2)*hypergeometric([1/2-n/2, 1-n/2], [3/2, 1/2-n, 1-n], -1/4)/sqrt(4*pi)
%o A053987 [round(A053987(n).n(100)) for n in (1..18)] # _Peter Luschny_, Sep 10 2014
%o A053987 (Magma)
%o A053987 A053987:= func< n| &+[(-1)^k*Factorial(2*n-2*k-1)/(Factorial(n-2*k-1)* Factorial(2*k+1)): k in [0..Floor((n-1)/2)]] >;
%o A053987 [A053987(n) : n in [1..20]]; // _G. C. Greubel_, May 17 2020
%Y A053987 Cf. A001517, A053988, A058798.
%K A053987 nonn,frac,easy
%O A053987 1,2
%A A053987 _Vladeta Jovovic_, Apr 03 2000
%E A053987 a(16)-a(17) from _Wesley Ivan Hurt_, Feb 28 2014