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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
1, 0, 8, 32, 2848, 87808, 97425920, 18364346368, 459757145081856, 468713931103109120, 349620381018764380930048, 1788712998645738038832398336, 46562065744123901943395531497144320 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..50

FORMULA

a(n) = ( C(2^n + n-1, n) + (-1)^n*C(2^n, n) )/2. - Paul D. Hanna, Nov 24 2009

EXAMPLE

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

The g.f. of A166996 is S(x):

S(x) = Sum_{n>=0} -log(1 - 2^(2n+1)*x)^(2n+1)/(2n+1)!

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

where C(x) + S(x) = Sum_{n>=0} C(2^n + n - 1, n)*x^n ... (cf. A060690)

and C(x) - S(x) = Sum_{n>=0} C(2^n, n)*(-x)^n ... (cf. A014070).

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 +...

MATHEMATICA

Table[(1/2)*(Binomial[2^n + n - 1, n ] + (-1)^n *Binomial[2^n, n]), {n, 0, 10}] (* G. C. Greubel, May 30 2016 *)

PROG

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

(PARI) {a(n)=(binomial(2^n + n-1, n) + (-1)^n*binomial(2^n, n))/2} \\ Paul D. Hanna, Nov 24 2009

CROSSREFS

Cf. A166996, A166997, A166998, A060690, A014070.

Sequence in context: A139306 A214594 A288457 * A079271 A336220 A247533

Adjacent sequences:  A166992 A166993 A166994 * A166996 A166997 A166998

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Nov 22 2009

STATUS

approved

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Last modified August 4 20:53 EDT 2021. Contains 346455 sequences. (Running on oeis4.)