A289069 Recurrence a(n+2) = Sum_{k=0..n} binomial(n,k)*a(k)*a(n+1-k) with a(0)=3, a(1)=-2.

3, -2, -6, -14, -6, 202, 1506, 4594, -29814, -573062, -4098606, 2741026, 487823034, 6657110122, 28995776706, -685482188846, -17937265077654, -181680546169382, 963087154054194, 72085899963332866, 1289184007236331674, 4679677879996688842, -383123191395931184094

Offset

0,1

Comments

One of a family of integer sequences whose e.g.f.s satisfy the differential equation f''(z) = f'(z)f(z). For more details, see A289064.

Links

Stanislav Sykora, Table of n, a(n) for n = 0..200

Stanislav Sykora, Sequences related to the differential equation f'' = af'f, Stan's Library, Vol. VI, Jun 2017.

Formula

E.g.f.: -sqrt(13)*tanh(z*sqrt(13)/2 - arccosh(sqrt(13)/2)).

E.g.f. for the sequence (-1)^(n+1)*a(n): -sqrt(13)*tanh(z*sqrt(13)/2 + arccosh(sqrt(13)/2)).

Mathematica

a[0] = 3; a[1] = -2; a[n_] := a[n] = Sum[Binomial[n - 2, k] a[k] a[n - k - 1], {k, 0, n - 2}]; Array[a, 23, 0] (* Michael De Vlieger, Jul 04 2017 *)

Prog

(PARI) c0=3; c1=-2; 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]));
  a
(Python)
from sympy import binomial
l=[3, -2]
for n in range(2, 51): l+=[sum(binomial(n - 2, k)*l[k]*l[n - 1 - k] for k in range(n - 1)), ]
print(l) # Indranil Ghosh, Jun 30 2017

Crossrefs

Sequences for other starting pairs: A000111 (1,1), A289064 (1,-1), A289065 (2,-1), A289066 (3,1), A289067 (3,-1), A289068 (1,-2), A289070 (0,3).

Sequence in context: A073883 A248982 A333446 * A074718 A285457 A007812

Adjacent sequences: A289066 A289067 A289068 * A289070 A289071 A289072

Keyword

sign

Author

Stanislav Sykora, Jun 23 2017

Status

approved