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%I #7 May 30 2016 19:15:39
%S 1,0,8,32,2848,87808,97425920,18364346368,459757145081856,
%T 468713931103109120,349620381018764380930048,
%U 1788712998645738038832398336,46562065744123901943395531497144320
%N G.f.: C(x) = Sum_{n>=0} log(1 - 2^(2n)*x)^(2n)/(2n)!, a power series in x with integer coefficients.
%H G. C. Greubel, <a href="/A166995/b166995.txt">Table of n, a(n) for n = 0..50</a>
%F a(n) = ( C(2^n + n-1, n) + (-1)^n*C(2^n, n) )/2. - _Paul D. Hanna_, Nov 24 2009
%e G.f: C(x) = 1 + 8*x^2 + 32*x^3 + 2848*x^4 + 87808*x^5 + 97425920*x^6 +...
%e The g.f. of A166996 is S(x):
%e S(x) = Sum_{n>=0} -log(1 - 2^(2n+1)*x)^(2n+1)/(2n+1)!
%e S(x) = 2*x + 2*x^2 + 88*x^3 + 1028*x^4 + 289184*x^5 + 22451552*x^6 +...
%e where C(x) + S(x) = Sum_{n>=0} C(2^n + n - 1, n)*x^n ... (cf. A060690)
%e and C(x) - S(x) = Sum_{n>=0} C(2^n, n)*(-x)^n ... (cf. A014070).
%e Related expansions:
%e C(x) + S(x) = 1 + 2*x + 10*x^2 + 120*x^3 + 3876*x^4 + 376992*x^5 +...
%e C(x) - S(x) = 1 - 2*x + 6*x^2 - 56*x^3 + 1820*x^4 - 201376*x^5 +...
%t Table[(1/2)*(Binomial[2^n + n - 1, n ] + (-1)^n *Binomial[2^n, n]), {n, 0, 10}] (* _G. C. Greubel_, May 30 2016 *)
%o (PARI) {a(n)=polcoeff(sum(k=0,n,log(1-2^(2*k)*x +x*O(x^n))^(2*k)/(2*k)!),n)}
%o (PARI) {a(n)=(binomial(2^n + n-1, n) + (-1)^n*binomial(2^n, n))/2} \\ _Paul D. Hanna_, Nov 24 2009
%Y Cf. A166996, A166997, A166998, A060690, A014070.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Nov 22 2009
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