A289082 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)=1, a(1)=i.

1, 0, 0, -1, -4, -11, -26, -23, 376, 4041, 28266, 153973, 517204, -1969651, -62317666, -796305583, -7467806224, -49765963119, -74804319534, 4766716058093, 107039955050444, 1549401029915989, 16531537283552134, 99639596341492297, -903257832030151064, -44705398895606766759

Offset

0,5

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

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/2-1/4) 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}]; Re[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]));
  real(a)

Crossrefs

Cf. A289083 (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), A289084 and A289085 (2,i), A289086 and A289087 (1,2i), A289088 and A289089 (2,2i).

Sequence in context: A328937 A328936 A266816 * A079467 A140897 A008263

Adjacent sequences: A289079 A289080 A289081 * A289083 A289084 A289085

Keyword

sign

Author

Stanislav Sykora, Jul 19 2017

Status

approved