A289084 Real 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.

2, 0, 0, -1, -8, -44, -208, -878, -2848, 1344, 165888, 2386288, 26058368, 242819936, 1859201152, 8317586192, -76925639168, -3065481255936, -60922964385792, -959095854652672, -12571364621944832, -124608931461592064, -359648778853101568, 25091103143957181952

Offset

0,1

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.: real(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}]; Re[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]));
  real(a)

Crossrefs

Cf. A289085 (imaginary 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: A109983 A332409 A367000 * A193033 A318253 A249772

Adjacent sequences: A289081 A289082 A289083 * A289085 A289086 A289087

Keyword

sign

Author

Stanislav Sykora, Jul 19 2017

Status

approved