This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A074790 #20 Jul 09 2021 11:23:53
%S A074790 1,5,101,4241,305353,33588829,5239857325,1100370038249,
%T A074790 299300650403729,102360822438075317,42991545423991633141,
%U A074790 21753721984539766369345,13052233190723859821607001,9162667699888149594768114701
%N A074790 a(n) = (2*n+1)!*Sum_{k=0..n} (-1)^k/(2*k+1)!.
%H A074790 Seiichi Manyama, <a href="/A074790/b074790.txt">Table of n, a(n) for n = 0..224</a>
%F A074790 a(n) = round(sin(1)*(2*n+1)!).
%F A074790 a(n) = A009551(2*n+1).
%F A074790 From _Peter Bala_, Jan 30 2015: (Start)
%F A074790 G.f.: sin(x)/(1 - x^2) = x + 5*x^3/3! + 101*x^5/5! + 4241*x^7/7! + ....
%F A074790 a(n) = 2*n*(2*n + 1)*a(n-1) + (-1)^n with a(0) = 1.
%F A074790 a(n) = (4*n^2 + 2*n - 1)*a(n-1) + (2*n-1)*(2*n-2)*a(n-2) with a(0) = 1, a(1) = 5.
%F A074790 The sequence b(n) := (2*n + 1)! also satisfies the second recurrence but with b(0) = 1, b(1) = 6. This leads to the continued fraction representation a(n) = (2*n + 1)!*(1 - 1/(6 + 6/(19 + 20/(41 + ... + (2*n - 1)*(2*n - 2)/(4*n^2 + 2*n - 1) )))) for n >= 2. Taking the limit gives the continued fraction representation sin(1) = 1 - 1/(6 + 6/(19 + 20/(41 + ... + (2*n - 1)*(2*n - 2)/((4*n^2 + 2*n - 1) + ... )))). (End)
%t A074790 Table[(2n+1)!Sum[(-1)^k/(2k+1)!,{k,0,n}],{n,0,20}] (* _Harvey P. Dale_, Sep 14 2019 *)
%o A074790 (PARI) a(n) = (2*n+1)!*sum(k=0,n,(-1)^k/(2*k+1)!); \\ _Michel Marcus_, Sep 09 2016
%o A074790 (Sage) [factorial(2*n+1)*sum((-1)^j/factorial(2*j+1) for j in (0..n)) for n in (0..20)] # _G. C. Greubel_, Jul 09 2021
%Y A074790 Cf. A000166, A009551, A049469, A051397.
%K A074790 nonn,easy
%O A074790 0,2
%A A074790 _Benoit Cloitre_, Sep 07 2002