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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 (list; graph; refs; listen; history; text; internal format)
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 be also complex integers. For more details, see A289064.

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/2-1/4) and L1 = acos(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 A206950 A189644

Adjacent sequences:  A289080 A289081 A289082 * A289084 A289085 A289086

KEYWORD

sign

AUTHOR

Stanislav Sykora, Jul 19 2017

STATUS

approved

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Last modified September 26 10:52 EDT 2017. Contains 292518 sequences.