A166996 G.f.: S(x) = Sum_{n>=0} -log(1 - 2^(2n+1)*x)^(2n+1)/(2n+1)!, a power series in x with integer coefficients.

2, 2, 88, 1028, 289184, 22451552, 112890141568, 50093449805856, 6676830881369059840, 15354513520142235310592, 66620888067382334066280699904, 750203718611121304644623635491840

Offset: 1

Author

Paul D. Hanna, Nov 22 2009

Keywords

• nonn

Examples

			G.f.: S(x) = 2*x + 2*x^2 + 88*x^3 + 1028*x^4 + 289184*x^5 + 22451552*x^6 + ...
The g.f. of A166995 is C(x):
C(x) = Sum_{n>=0} log(1 - 2^(2n)*x)^(2n)/(2n)!.
C(x) = 1 + 8*x^2 + 32*x^3 + 2848*x^4 + 87808*x^5 + 97425920*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 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..50

Crossrefs

Cf. A166995, A166997, A166998, A060690, A014070.

Programs

Mathematica

Table[(1/2)*(Binomial[2^n + n - 1, n ] - (-1)^n *Binomial[2^n, n]), {n, 50}] (* G. C. Greubel, May 30 2016 *)

PARI

{a(n)=polcoeff(-sum(k=0,n,log(1-2^(2*k+1)*x +x*O(x^n))^(2*k+1)/(2*k+1)!),n)}

PARI

{a(n)=(binomial(2^n + n-1, n) - (-1)^n*binomial(2^n, n))/2} \\ Paul D. Hanna, Nov 24 2009

Formula

a(n) = (binomial(2^n + n-1, n) - (-1)^n*binomial(2^n, n) )/2. [Paul D. Hanna, Nov 24 2009]