OFFSET
0,5
COMMENTS
One of a family of integer 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. When the first term changes sign, the effect is the inversion of the signs of all even terms, leaving all absolute values intact. There are many related sequences in the OEIS (see the link). Starting with a(0) = a(1) = 1, for example, one obtains A000111. All such sequences have a well defined, explicit e.g.f. (see the link).
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(3)*tanh(z*sqrt(3)/2 - arccosh(sqrt(3/2))).
E.g.f. for the same sequence, but with inverted signs of even terms: -sqrt(3)*tanh(z*sqrt(3)/2 + arccosh(sqrt(3/2))).
MATHEMATICA
a[n_] := a[n] = Sum[Binomial[n-2, k]*a[k]*a[n-k-1], {k, 0, n-2}]; a[0] = 1; a[1] = -1; Array[a, 26, 0] (* Jean-François Alcover, Jul 20 2017 *)
PROG
(PARI) c0=1; c1=-1; nmax = 200; \\ Initialize
a=vector(nmax+1)); a[1]=c0; a[2]=c1; \\ Compute
for(m=0, #a-3, a[m+3]=sum(k=0, m, binomial(m, k)*a[k+1]*a[m+2-k]));
a \\ Display
CROSSREFS
KEYWORD
sign
AUTHOR
Stanislav Sykora, Jun 23 2017
STATUS
approved