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A009744 Expansion of e.g.f. tan(x)*sin(x) (even powers only).

%I A009744 #42 Jun 13 2021 16:17:00
%S A009744 0,2,4,62,1384,50522,2702764,199360982,19391512144,2404879675442,
%T A009744 370371188237524,69348874393137902,15514534163557086904,
%U A009744 4087072509293123892362,1252259641403629865468284,441543893249023104553682822,177519391579539289436664789664
%N A009744 Expansion of e.g.f. tan(x)*sin(x) (even powers only).
%H A009744 T. D. Noe, <a href="/A009744/b009744.txt">Table of n, a(n) for n = 0..100</a>
%F A009744 G.f.: 1/G(0) - 1/(1+x) where G(k) = 1 - x*(2*k+1)^2/(1 - x*(2*k+2)^2/G(k+1) ); (recursively defined continued fraction). - _Sergei N. Gladkovskii_, Feb 05 2013
%F A009744 G.f.: 1/G(0) - 1/(1+x) where G(k) = 1 - x*(k+1)^2/G(k+1); (recursively defined continued fraction). - _Sergei N. Gladkovskii_, Feb 09 2013
%F A009744 a(n) ~ (2*n)! * 4^(n+1) / Pi^(2*n+1). - _Vaclav Kotesovec_, Jan 24 2015
%F A009744 Conjectural o.g.f.: Sum_{n >= 0} 4*x/2^n * Sum_{k = 0..n} (-1)^k*(k+1)*binomial(n,k)/( (1 + x*(2*k + 1)^2)*(1 + x*(2*k + 3)^2) ) = 2*x + 4*x^2 + 62*x^3 + 1384*x^4 + .... - _Peter Bala_, Mar 03 2015
%F A009744 From _Peter Luschny_, Jun 13 2021: (Start)
%F A009744 a(n) = (-1)^n*(Euler(2*n) - 1).
%F A009744 a(n) ~ 4^(2*n + 3/2)*exp(1/(24*n) - 2*n)*(n/Pi)^(2*n + 1/2). (End)
%p A009744 seq((2*i)!*coeff(series(tan(x)*sin(x),x,30),x,2*i),i=0..14); # _Peter Luschny_, Jul 14 2012
%t A009744 nn = 30; t = Range[0, nn]! CoefficientList[Series[Tan[x]*Sin[x], {x, 0, nn}], x]; Take[t, {1, nn, 2}] (* _T. D. Noe_, Jul 15 2012 *)
%o A009744 (Sage) # Variant of an algorithm of L. Seidel (1877) with a(0) = 1.
%o A009744 def A009744_list(n) :
%o A009744     dim = 2*n; E = matrix(ZZ, dim); E[0, 0] = 1
%o A009744     for m in (1..dim-1) :
%o A009744         if m % 2 == 0 :
%o A009744             E[m, 0] = 1;
%o A009744             for k in range(m-1, -1, -1) :
%o A009744                 E[k, m-k] = E[k+1, m-k-1] - E[k, m-k-1]
%o A009744         else :
%o A009744             E[0, m] = 1;
%o A009744             for k in range(1, m+1, 1) :
%o A009744                 E[k, m-k] = E[k-1, m-k+1] + E[k-1, m-k]
%o A009744     return [(-1)^(k//2)*E[0,k] for k in range(dim) if is_even(k)]
%o A009744 A009744_list(14)  # _Peter Luschny_, Jul 14 2012
%o A009744 (PARI) x='x+O('x^50); v=Vec(serlaplace(tan(x)*sin(x))); concat([0], vector(#v\2,n,v[2*n-1])) \\ _G. C. Greubel_, Mar 04 2018
%Y A009744 Cf. A029582, A099023.
%K A009744 nonn,easy
%O A009744 0,2
%A A009744 _R. H. Hardin_
%E A009744 Extended and signs tested by _Olivier Gérard_, Mar 15 1997