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
KEYWORD
sign
AUTHOR
Stanislav Sykora, Jun 23 2017
STATUS
approved