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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.

%I A166996 #16 Apr 07 2020 03:35:40
%S A166996 2,2,88,1028,289184,22451552,112890141568,50093449805856,
%T A166996 6676830881369059840,15354513520142235310592,
%U A166996 66620888067382334066280699904,750203718611121304644623635491840
%N 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.
%H A166996 G. C. Greubel, <a href="/A166996/b166996.txt">Table of n, a(n) for n = 1..50</a>
%F A166996 a(n) = (binomial(2^n + n-1, n) - (-1)^n*binomial(2^n, n) )/2. [_Paul D. Hanna_, Nov 24 2009]
%e A166996 G.f.: S(x) = 2*x + 2*x^2 + 88*x^3 + 1028*x^4 + 289184*x^5 + 22451552*x^6 + ...
%e A166996 The g.f. of A166995 is C(x):
%e A166996 C(x) = Sum_{n>=0} log(1 - 2^(2n)*x)^(2n)/(2n)!.
%e A166996 C(x) = 1 + 8*x^2 + 32*x^3 + 2848*x^4 + 87808*x^5 + 97425920*x^6 + ...
%e A166996 where C(x) + S(x) = Sum_{n>=0} C(2^n + n - 1, n)*x^n ... (cf. A060690)
%e A166996 and C(x) - S(x) = Sum_{n>=0} C(2^n, n)*(-x)^n ... (cf. A014070).
%e A166996 Related expansions:
%e A166996 C(x) + S(x) = 1 + 2*x + 10*x^2 + 120*x^3 + 3876*x^4 + 376992*x^5 + ...
%e A166996 C(x) - S(x) = 1 - 2*x + 6*x^2 - 56*x^3 + 1820*x^4 - 201376*x^5 + ...
%t A166996 Table[(1/2)*(Binomial[2^n + n - 1, n ] - (-1)^n *Binomial[2^n, n]), {n, 50}] (* _G. C. Greubel_, May 30 2016 *)
%o A166996 (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)}
%o A166996 (PARI) {a(n)=(binomial(2^n + n-1, n) - (-1)^n*binomial(2^n, n))/2} \\ _Paul D. Hanna_, Nov 24 2009
%Y A166996 Cf. A166995, A166997, A166998, A060690, A014070.
%K A166996 nonn
%O A166996 1,1
%A A166996 _Paul D. Hanna_, Nov 22 2009