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A166995 G.f.: C(x) = Sum_{n>=0} log(1 - 2^(2n)*x)^(2n)/(2n)!, a power series in x with integer coefficients. 4

%I

%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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Last modified September 22 21:04 EDT 2021. Contains 347608 sequences. (Running on oeis4.)