OFFSET
0,4
COMMENTS
Second binomial transform of the expansion of c(-x)^3. The g.f. is transformed to 1-x under the Chebyshev transformation A(x)->(1/(1+x^2))A(x/(1+x^2)).
A208355(n) = abs(a(n)). - Reinhard Zumkeller, Mar 03 2012
FORMULA
G.f.: (1-(1-x)c(x^2))/x where c(x) is the g.f. of the Catalan numbers A000108.
a(n) = sum{k=0..n, (k+1)C(n, (n-k)/2)(0^k-sum{j=0..k, C(k, j)(-1)^(k-j)*j})(1+(-1)^(n-k))/(n+k+2)}.
a(n) = (-1)^n A208355(n) = (-1)^n A000108([(n+1)/2]): Repeated Catalan numbers with alternating sign. - M. F. Hasler, Aug 25 2012
Conjecture: (n+3)*a(n) +(-n-1)*a(n-1) -4*n*a(n-2) +4*(n-2)*a(n-3)=0. - R. J. Mathar, Nov 26 2012
MATHEMATICA
Table[(-1)^n CatalanNumber[Floor[(n+1)/2]], {n, 0, 38}] (* Jean-François Alcover, Jun 11 2019 *)
PROG
(Sage)
def A099363_list(n) :
D = [0]*(n+2); D[1] = 1
b = True; h = 2; R = []
for i in range(2*n-1) :
if b :
for k in range(h, 0, -1) : D[k] -= D[k-1]
h += 1; R.append((-1)^(h//2)*D[2])
else :
for k in range(1, h, 1) : D[k] += D[k+1]
b = not b
return R
A099363_list(39) # Peter Luschny, Jun 03 2012
CROSSREFS
KEYWORD
easy,sign
AUTHOR
Paul Barry, Oct 13 2004
STATUS
approved