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A289083
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)=1, a(1)=i.
8
0, 1, 1, 1, 1, -3, -33, -179, -767, -2407, 863, 107489, 1261697, 10505157, 65544687, 192284981, -2621700607, -64255381967, -880236943937, -8946701130551, -59459763005183, 104310663206877, 12346394790353487, 268724181711473821, 4054748296390774273
OFFSET
0,6
COMMENTS
Here, i is the imaginary unit. The complex integer sequence c(n) = A289082(n) + i*A289083(n) is one of a family of sequences whose e.g.f.s satisfy the differential equation f''(z) = f'(z)f(z). Each such sequence is uniquely characterized by its two starting terms, which may also be complex integers. For more details, see A289064.
LINKS
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/2-1/4) and L1 = arccos(sqrt(i/2+1)).
MATHEMATICA
a[0]=1; a[1]=I; a[n_]:=a[n]=Sum[Binomial[n - 2, k] a[k] a[n - 1 - k], {k, 0, n - 2}]; Im[Table[a[n], {n, 0, 50}]] (* Indranil Ghosh, Jul 20 2017 *)
PROG
(PARI) c0=1; c1=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. A289082 (real part).
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), A289084 and A289085 (2,i), A289086 and A289087 (1,2i), A289088 and A289089 (2,2i).
Sequence in context: A279885 A132122 A203560 * A195578 A370083 A331193
KEYWORD
sign
AUTHOR
Stanislav Sykora, Jul 19 2017
STATUS
approved