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

1,1

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

Table of n, a(n) for n=1..16.

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

Cf. A030178, A370491 (numerator).

Sequence in context: A373870 A070813 A156214 * A187654 A280517 A354511

Adjacent sequences: A370487 A370488 A370489 * A370491 A370492 A370493

Keyword

nonn

Author

Raul Prisacariu, Feb 19 2024

Extensions

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

Status

approved