OFFSET
1,1
COMMENTS
Whittaker's root series formula is applied to 1 - 2x + x^2/2! - x^3/3! + x^4/4! - x^5/5! + x^6/6! - ..., which is the Taylor expansion of -x + e^(-x). We obtain the following infinite series that converges to the Omega constant (LambertW(1)): LambertW(1) = 1/2 + 1/14 - 1/259 - 5/9657 + 19/200187 - 3/18671081 ... . The sequence is formed by the denominators of the infinite series.
LINKS
E. T. Whittaker and G. Robinson, The Calculus of Observations, London: Blackie & Son, Ltd. 1924, pp. 120-123.
FORMULA
for n>1, a(n) is the denominator of the simplified fraction -det ToeplitzMatrix((c(2),c(1),c(0),0,0,...,0),(c(2),c(3),c(4),...,c(n)))/(det ToeplitzMatrix((c(1),c(0),0,...,0),(c(1),c(2),c(3),...,c(n-1)))*det ToeplitzMatrix((c(1),c(0),0,...,0),(c(1),c(2),c(3),...,c(n)))), where c(0)=1, c(1)=-2, c(n) = (-1)^n/n!.
EXAMPLE
a(1) is the denominator of -1/-2 = 1/2.
a(2) is the denominator of -(1/2)/((-2)*det ToeplitzMatrix((-2,1),(-2,1/2!)) = -(1/2)/((-2)(7/2)) = 1/14.
a(3) is the denominator of -det ToeplitzMatrix((1/2!,-2),(1/2!,-1/3!))/(det ToeplitzMatrix((-2,1),(-2,1/2!)*det ToeplitzMatrix((-2,1,0),(-2,1/2!,-1/3!))) = -(-1/12)/((7/2)(-37/6)) = -1/259.
CROSSREFS
KEYWORD
nonn
AUTHOR
Raul Prisacariu, Feb 19 2024
EXTENSIONS
a(9)-a(16) from Chai Wah Wu, Mar 23 2024
STATUS
approved