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%I #23 Sep 08 2022 08:45:00
%S 2,11,108,1501,26910,590519,15326584,459207001,15597711450,
%T 592253828099,24859063068708,1142924647332469,57121373303554742,
%U 3083411233744623599,178780730183884614000,11081321860167101444401,731188462040844810716466,51172111020998969648708219
%N Denominators 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 G. C. Greubel, <a href="/A053988/b053988.txt">Table of n, a(n) for n = 1..360</a>
%F a(n) = Sum_{k=0..floor(n/2)} (-1)^k*(2*n-2*k)!/((n-2*k)!*(2*k)!) - _Benoit Cloitre_, Jan 03 2006
%F From _G. C. Greubel_, May 13 2020: (Start)
%F E.g.f.: cos((1 - sqrt(1-4*x))/2)/sqrt(1-4*x) - 1.
%F a(n) = 2*(2*n-1)*a(n-1) - a(n-2).
%F a(n) = ((-i)^n/2)*(y(n, 2*i) + (-1)^n*y(n, -2*i)), where y(n, x) are the Bessel Polynomials. (End)
%F a(n) ~ cos(1/2) * 2^(2*n + 1/2) * n^n / exp(n). - _Vaclav Kotesovec_, May 14 2020
%p a:= n -> add((-1)^k*(2*n-2*k)!/((n-2*k)!*(2*k)!), k = 0..floor(n/2));
%p seq(a(n), n = 1..20); # _G. C. Greubel_, May 13 2020
%t Table[Sum[(-1)^k*(2*n-2*k)!/((n-2*k)!*(2*k)!), {k,0,Floor[n/2]}], {n, 20}] (* _G. C. Greubel_, May 13 2020 *)
%o (PARI) a(n)=sum(k=0,floor(n/2),(-1)^k*(2*n-2*k)!/(n-2*k)!/(2*k)!) \\ _Benoit Cloitre_, Jan 03 2006
%o (Magma) [(&+[ (-1)^k*Factorial(2*n-2*k)/(Factorial(n-2*k)*Factorial(2*k)): k in [0..Floor(n/2)]] ): n in [1..20]]; // _G. C. Greubel_, May 13 2020
%o (Sage) [sum((-1)^k*factorial(2*n-2*k)/(factorial(n-2*k)*factorial(2*k)) for k in (0..floor(n/2))) for n in (1..20)] # _G. C. Greubel_, May 13 2020
%Y Cf. A001497, A053987 (numerators), A161011 (tan(1/2)).
%K easy,frac,nonn
%O 1,1
%A _Vladeta Jovovic_, Apr 03 2000
%E More terms from _G. C. Greubel_, May 13 2020
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