A049281 Numerators of coefficients in power series for -log(1+x)*log(1-x).

1, 5, 47, 319, 1879, 20417, 263111, 52279, 1768477, 33464927, 166770367, 3825136961, 19081066231, 57128792093, 236266661971, 7313175618421, 14606816124167, 102126365345729, 3774664307989373, 3771059091081773

Offset

1,2

Links

Muniru A Asiru, Table of n, a(n) for n = 1..300

Formula

a(n)/A069685(n) = Integral_{x=0..1} x^(n-1)*log(1 + 1/sqrt(x)) dx = 1/n*Sum_{k=0..2*n-2} (-1)^k/(2*n-1-k).

From Peter Bala, Feb 21 2017: (Start)

a(n) = numerator((1/n)*Sum_{k=1..2*n-1} (-1)^(k-1)/k ). Cf. A058313.

a(n) = numerator((1/2)*binomial(2*n,n)*Sum_{k=0..n-1} (-1)^k* binomial(n-1,k)/(n + k)^2 ).

The coefficients in the expansion of log(1 + x)*log(1 - x) are given by (1/2)*binomial(2*n,n)*Integral_{x = 0..1} (x*(1 - x))^(n-1)*log(x) dx.

log(1 + x)*log(1 - x) = (1/2)*Integral_{z = 0..1} log(z)/(z*(1 - z)) * (1/sqrt( 1 - 4*x^2*z*(1 - z) ) - 1) dz. (End)

Example

-log(1 + x)*log(1 - x) = x^2 + 5/12*x^4 + 47/180*x^6 + 319/1680*x^8 + ...

Maple

seq(numer(1/n*add( (-1)^k/(2*n-1-k), k = 0..2*n-2)), n = 1..20); # Peter Bala, Feb 21 2017

Prog

(GAP) List(List([1..25], n->(1/n)*Sum([1..2*n-1], k->(-1)^(k-1)/k)), NumeratorRat); # Muniru A Asiru, Jun 01 2018

Crossrefs

Bisection of A058313. Cf. A069685 (denominators).

Sequence in context: A352303 A344121 A304371 * A354894 A219073 A327595

Adjacent sequences: A049278 A049279 A049280 * A049282 A049283 A049284

Keyword

easy,frac,nonn

Author

Benoit Cloitre, May 03 2002

Status

approved