OFFSET
0,2
COMMENTS
More generally, given {S} such that: S(n) = b*S(n-1) + c*S(n-2), |b|>0, |c|>0, S(0)=1, then Sum_{n>=0} S(n)*Catalan(n)*x^n = sqrt( (1-2*b*x - sqrt(1-4*b*x-16*c*x^2))/(2*b^2+8*c) )/x.
Conjecture: +n*(n+1)*a(n) -4*n*(2*n-1)*a(n-1) -8*(2*n-1)*(2*n-3)*a(n-2)=0. - R. J. Mathar, Oct 08 2016
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..960
FORMULA
G.f.: sqrt( (1-4*x - sqrt(1-8*x-32*x^2))/24 )/x.
EXAMPLE
MATHEMATICA
MapIndexed[CatalanNumber[#2 - 1] #1 &, Rest@ RecurrenceTable[{a[n] == 2 (a[n - 1] + a[n - 2]), a[0] == 0, a[1] == 1}, a, {n, 22}]] // Flatten (* or *)
CoefficientList[Series[Sqrt[(1 - 4 x - Sqrt[1 - 8 x - 32 x^2])/24]/x, {x, 0, 21}], x] (* Michael De Vlieger, Oct 08 2016 *)
PROG
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 26 2012
STATUS
approved