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.

1, 0, 6, 28, 2684, 85664, 96848424, 18318978896, 459531493100736, 468613553577122688, 349607028167776160389536, 1788682277200384090414421312, 46561932503015793339090359576558496

Offset

0,3

Links

Table of n, a(n) for n=0..12.

Formula

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.

Self-convolution yields A166998.

Example

G.f: 1 + 6*x^2 + 28*x^3 + 2684*x^4 + 85664*x^5 + 96848424*x^6 +...
which equals sqrt( C(x)^2 - S(x)^2 ) where
C(x) = 1 + 8*x^2 + 32*x^3 + 2848*x^4 + 87808*x^5 + 97425920*x^6 +...
S(x) = 2*x + 2*x^2 + 88*x^3 + 1028*x^4 + 289184*x^5 + 22451552*x^6 +...
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 +...

Prog

(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)}

Crossrefs

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

Sequence in context: A066239 A097464 A354072 * A038182 A095723 A057246

Adjacent sequences: A166995 A166996 A166997 * A166999 A167000 A167001

Keyword

nonn

Author

Paul D. Hanna, Nov 22 2009

Status

approved