A289085 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)=i.

0, 1, 2, 4, 8, 12, -36, -656, -6016, -45712, -303584, -1614784, -3151424, 89449152, 2020752864, 30106674944, 371094759424, 3803441275648, 27086999353856, -53440551394304, -7360216885479424, -195653223115035648, -3852848368364645376, -62648371228429684736

Offset

0,3

Comments

Here, i is the imaginary unit. The complex integer sequence c(n) = A289084(n) + i*A289085(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/2-1) and L1 = arccos(sqrt(1+2*i)).

Mathematica

a[0]=2; 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=2; 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. A289084 (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), A289082 and A289083 (1,i), A289086 and A289087 (1,2i), A289088 and A289089 (2,2i).

Sequence in context: A202148 A215825 A177268 * A242924 A133802 A076202

Adjacent sequences: A289082 A289083 A289084 * A289086 A289087 A289088

Keyword

sign

Author

Stanislav Sykora, Jul 19 2017

Status

approved