|
%I #18 Aug 18 2023 23:36:26
%S 1,0,0,-4,-16,-44,-104,316,7456,67620,458280,1945036,-8476496,
%T -329703244,-4937610184,-52376610724,-339229127104,1480668009540,
%U 98702565516360,2094684947562476,30513682733622224,275602192570387156,-985849997324785064,-115940732145525976964
%N Real parts of the recursive sequence a(n+2) = Sum_{k=0..n} binomial(n,k)*a(k)*a(n+1-k), with a(0)=1, a(1)=2i.
%C Here, i is the imaginary unit. The complex integer sequence c(n) = A289086(n) + i*A289087(n) is one of a family of sequences whose e.g.f.s satisfy the differential equation f''(z) = f'(z)f(z). For more details, see A289064 and A289082.
%H Stanislav Sykora, <a href="/A289086/b289086.txt">Table of n, a(n) for n = 0..200</a>
%H S. Sykora, <a href="http://dx.doi.org/10.3247/SL6Math17.001">Sequences related to the differential equation f'' = af'f</a>, Stan's Library, Vol. VI, Jun 2017.
%F E.g.f.: real(2*L0*tan(L0*z + L1)), where L0 = sqrt(i-1/4) and L1 = arccos(sqrt(1+i/4)).
%t a[0]=1; a[1]=2I; a[n_]:=a[n]=Sum[Binomial[n - 2, k] a[k] a[n - 1 - k], {k, 0, n - 2}]; Re[Table[a[n], {n, 0, 50}]] (* _Indranil Ghosh_, Jul 20 2017 *)
%o (PARI) c0=1; c1=2*I; nmax = 200;
%o a=vector(nmax+1); a[1]=c0; a[2]=c1;
%o for(m=0, #a-3, a[m+3]=sum(k=0, m, binomial(m, k)*a[k+1]*a[m+2-k]));
%o real(a)
%Y Cf. A289087 (imaginary part).
%Y Sequences for other starting pairs: A000111 (1,1), A289064 (1,-1), A289065 (2,-1), A289066 (3,1), A289067 (3,-1), A289068 (1,-2), A289069 (3,-2), A289070 (0,3), A289082 and A289083 (1,i), A289084 and A289085 (2,i), A289088 and A289089 (2,2i).
%K sign
%O 0,4
%A _Stanislav Sykora_, Jul 19 2017
|
