A220113 - OEIS

%I #16 Feb 22 2013 07:14:11

%S 1,-2,-12,-424,-29808,-2966816,-237449920,76118167936,84317834342656,

%T 53499781544238592,20080969948883956736,-10740526073453596649472,

%U -31099457241702481710116864

%N E.g.f. A(x)=sum{n>0, a(n)x^(2*n-1)/(2*n-1)!} satisfies A(A(x))=sin(2*x)/2.

%H Dmitry Kruchinin, Vladimir Kruchinin, <a href="http://arxiv.org/abs/1302.1986">Method for solving an iterative functional equation $A^{2^n}(x)=F(x)$</a>, arXiv:1302.1986

%F a(n)=T(2*n-1,1), T(n,m)=1/2*(2^(n-2*m)*(((-1)^(n-m)+1)*sum(i=0..m/2, (2*i-m)^n*binomial(m,i)*(-1)^((n+m)/2-i)))/m!-sum(i=m+1..n-1, T(n,i)*T(i,m))), n>m, T(n,n)=1.

%t t[n_, m_] := t[n, m] = 1/2*(2^(n - 2*m)*(((-1)^(n-m) + 1)* Sum[(2*i - m)^n*Binomial[m, i]*(-1)^((n+m)/2 - i), {i, 0, m/2}])/m! - Sum[t[n, i]*t[i, m], {i, m+1, n-1}]); t[n_, n_] = 1; Table[ t[2*n-1, 1], {n, 1, 13}] (* _Jean-François Alcover_, Feb 22 2013 *)

%o (Maxima)

%o T(n,m):=if n=m then 1 else 1/2*(2^(n-2*m)*(((-1)^(n-m)+1)*sum((2*i-m)^n*binomial(m,i)*(-1)^((n+m)/2-i),i,0,m/2))/m!-sum(T(n,i)*T(i,m),i,m+1,n-1));

%o makelist(((T3(2*n-1,1))),n,1,7);

%Y Cf. A048602.

%K sign

%O 1,2

%A _Dmitry Kruchinin_, Dec 05 2012