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A166998 G.f.: sqrt(C(x)^2 - S(x)^2) where C(x) = Sum_{n>=0} log(1 - 2^(2n)*x)^(2n)/(2n)! and S(x) = Sum_{n>=0} -log(1 - 2^(2n+1)*x)^(2n+1)/(2n+1)! are the g.f.s of A166995 and A166996, respectively. 4

%I

%S 1,0,6,28,2684,85664,96848424,18318978896,459531493100736,

%T 468613553577122688,349607028167776160389536,

%U 1788682277200384090414421312,46561932503015793339090359576558496

%N G.f.: sqrt(C(x)^2 - S(x)^2) where C(x) = Sum_{n>=0} log(1 - 2^(2n)*x)^(2n)/(2n)! and S(x) = Sum_{n>=0} -log(1 - 2^(2n+1)*x)^(2n+1)/(2n+1)! are the g.f.s of A166995 and A166996, respectively.

%F G.f.: sqrt([C(x)+S(x)]*[C(x)-S(x)]) where C(x) + S(x) = g.f. of A060690 and C(-x) - S(-x) = g.f. of A014070.

%F Self-convolution yields A166998.

%e G.f: 1 + 6*x^2 + 28*x^3 + 2684*x^4 + 85664*x^5 + 96848424*x^6 +...

%e which equals sqrt( C(x)^2 - S(x)^2 ) where

%e C(x) = 1 + 8*x^2 + 32*x^3 + 2848*x^4 + 87808*x^5 + 97425920*x^6 +...

%e S(x) = 2*x + 2*x^2 + 88*x^3 + 1028*x^4 + 289184*x^5 + 22451552*x^6 +...

%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 +...

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

%Y Cf. A166995, A166996, A166997, A060690, A014070.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Nov 22 2009

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Last modified July 27 05:26 EDT 2021. Contains 346305 sequences. (Running on oeis4.)