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Imaginary parts of the recursive sequence a(n+2) = Sum_{k=0..n} binomial(n,k)*a(k)*a(n+1-k), with a(0)=2, a(1)=2i.
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%I #8 Aug 18 2023 23:36:41

%S 0,2,4,8,16,0,-480,-5632,-48896,-355328,-1901056,667648,238217216,

%T 4976025600,75193896960,911384117248,7730931236864,-6549305294848,

%U -2307879895433216,-69748749748928512,-1498587541480669184,-25307341803434803200,-292113960612790272000

%N Imaginary parts of the recursive sequence a(n+2) = Sum_{k=0..n} binomial(n,k)*a(k)*a(n+1-k), with a(0)=2, a(1)=2i.

%C Here, i is the imaginary unit. The complex integer sequence c(n) = A289088(n) + i*A289089(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="/A289089/b289089.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.: imag(2*L0*tan(L0*z + L1)), where L0 = sqrt(i-1) and L1 = arccos(sqrt(i+1)).

%o (PARI) c0=2; 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 imag(a)

%Y Cf. A289088 (real 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), A289086 and A289087 (1,2i).

%K sign

%O 0,2

%A _Stanislav Sykora_, Jul 24 2017