A370490 The denominators of a series that converges to the Omega constant (A030178) obtained using Whittaker's root series formula.

2, 14, 259, 9657, 200187, 18671081, 7313976065, 1273374259615, 285038137030769, 79755360301275363, 9091712937155442435, 149243024021521700285, 1085736156475373087072485, 3071709182054627484879798019, 2005459027715242401528647218817, 1496371535371115486607560677791759

Offset: 1

Author

Raul Prisacariu, Feb 19 2024

Keywords

• nonn

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.

Examples

			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.
		

Links

• E. T. Whittaker and G. Robinson, The Calculus of Observations, London: Blackie & Son, Ltd. 1924, pp. 120-123.

Crossrefs

Cf. A030178, A370491 (numerator).

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!.

Extensions

a(9)-a(16) from Chai Wah Wu, Mar 23 2024