A289088 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)=2i.

2, 0, 0, -4, -32, -176, -832, -3104, 512, 211200, 3425280, 40761856, 395890688, 2742339584, -1576804352, -553831911424, -14161881202688, -259252051968000, -3761903248343040, -37698142004445184, 44198204416196608, 15672885673387884544, 545945384701738876928

Offset

0,1

Comments

Here, i is the imaginary unit. The complex integer sequence c(n) = A289088(n) + i*A289089(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-1) and L1 = arccos(sqrt(i+1)).

Mathematica

a[n_] := a[n] = Which[n == 0, 2, n == 1, 2 I, True, Sum[Binomial[n - 2, k] a[k] a[n - 1 - k], {k, 0, n - 2}]]; Re@ Array[a, 23, 0] (* Michael De Vlieger, Jul 20 2017 *)

Prog

(PARI) c0=2; c1=2*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. A289089 (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), A289084 and A289085 (2,i), A289086 and A289087 (1,2i).

Sequence in context: A159814 A169774 A302689 * A057611 A329959 A259701

Adjacent sequences: A289085 A289086 A289087 * A289089 A289090 A289091

Keyword

sign

Author

Stanislav Sykora, Jul 19 2017

Status

approved