OFFSET
0,3
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..50
FORMULA
a(n) = ( C(2^n + n-1, n) + (-1)^n*C(2^n, n) )/2. - Paul D. Hanna, Nov 24 2009
EXAMPLE
G.f: C(x) = 1 + 8*x^2 + 32*x^3 + 2848*x^4 + 87808*x^5 + 97425920*x^6 +...
The g.f. of A166996 is S(x):
S(x) = Sum_{n>=0} -log(1 - 2^(2n+1)*x)^(2n+1)/(2n+1)!
S(x) = 2*x + 2*x^2 + 88*x^3 + 1028*x^4 + 289184*x^5 + 22451552*x^6 +...
where C(x) + S(x) = Sum_{n>=0} C(2^n + n - 1, n)*x^n ... (cf. A060690)
and C(x) - S(x) = Sum_{n>=0} C(2^n, n)*(-x)^n ... (cf. A014070).
Related expansions:
C(x) + S(x) = 1 + 2*x + 10*x^2 + 120*x^3 + 3876*x^4 + 376992*x^5 +...
C(x) - S(x) = 1 - 2*x + 6*x^2 - 56*x^3 + 1820*x^4 - 201376*x^5 +...
MATHEMATICA
Table[(1/2)*(Binomial[2^n + n - 1, n ] + (-1)^n *Binomial[2^n, n]), {n, 0, 10}] (* G. C. Greubel, May 30 2016 *)
PROG
(PARI) {a(n)=polcoeff(sum(k=0, n, log(1-2^(2*k)*x +x*O(x^n))^(2*k)/(2*k)!), n)}
(PARI) {a(n)=(binomial(2^n + n-1, n) + (-1)^n*binomial(2^n, n))/2} \\ Paul D. Hanna, Nov 24 2009
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Nov 22 2009
STATUS
approved