A289089 - OEIS

A289089

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.

0, 2, 4, 8, 16, 0, -480, -5632, -48896, -355328, -1901056, 667648, 238217216, 4976025600, 75193896960, 911384117248, 7730931236864, -6549305294848, -2307879895433216, -69748749748928512, -1498587541480669184, -25307341803434803200, -292113960612790272000

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OFFSET

0,2

COMMENTS

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.

LINKS

Stanislav Sykora, Table of n, a(n) for n = 0..200

S. Sykora, Sequences related to the differential equation f'' = af'f, Stan's Library, Vol. VI, Jun 2017.

FORMULA

E.g.f.: imag(2*L0*tan(L0*z + L1)), where L0 = sqrt(i-1) and L1 = arccos(sqrt(i+1)).

PROG

(PARI) c0=2; c1=2*I; nmax = 200;

a=vector(nmax+1); a[1]=c0; a[2]=c1;

for(m=0, #a-3, a[m+3]=sum(k=0, m, binomial(m, k)*a[k+1]*a[m+2-k]));

imag(a)

CROSSREFS

Cf. A289088 (real part).