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
S. Sykora, Sequences related to the differential equation f'' = af'f, Stan's Library, Vol. VI, Jun 2017.
FORMULA
E.g.f.: -sqrt(7)/tanh(z*sqrt(7)/2 - arccosh(3/sqrt(2))).
E.g.f. for the sequence (-1)^(n+1)*a(n): -sqrt(7)/tanh(z*sqrt(7)/2 + arccosh(3/sqrt(2))).
a(n) ~ 2 * n! * 7^((n+1)/2) / log(8 + 3*sqrt(7))^(n+1). - Vaclav Kotesovec, Jun 24 2017
MAPLE
f:= proc(n) option remember; add(binomial(n-2, k)*procname(k)*procname(n-1-k), k=0..n-2) end proc:
f(0):= 3: f(1):= 1:
map(f, [$0..50]); # Robert Israel, Jul 20 2017
MATHEMATICA
a[n_] := a[n] = Sum[Binomial[n-2, k]*a[k]*a[n-k-1], {k, 0, n-2}]; a[0] = 3; a[1] = 1; Array[a, 23, 0] (* Jean-François Alcover, Jul 20 2017 *)
PROG
(PARI) c0=3; c1=1; 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
CROSSREFS
KEYWORD
nonn
AUTHOR
Stanislav Sykora, Jun 23 2017
STATUS
approved