OFFSET
0,3
COMMENTS
Let C(n) = 4^n*Gamma(n+1/2)/(sqrt(Pi)*Gamma(n+2)), then C'(n) = C(n)*(H(n-1/2) - H(n+1) + log(4)), where H(n) = Sum_{k>=1} (1/k-1/(n+k)) are harmonic numbers.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
FORMULA
a(n) = numerator(d(n)), where d(n) satisfies recurrence: d(0) = -1, d(1) = 1/2, (n+1)^2*d(n) = 2*(4*n^2-2*n-1)*d(n-1) - 4*(2*n-3)^2*d(n-2).
EXAMPLE
For n = 3, C'(3) = 59/12, so a(3) = numerator(59/12) = 59.
MATHEMATICA
Numerator@FunctionExpand@Table[CatalanNumber'[n] , {n, 0, 22}]
CROSSREFS
KEYWORD
sign,frac
AUTHOR
_Vladimir Reshetnikov_, Nov 11 2015
STATUS
approved