A370491 The numerators of a series that converges to the Omega constant (A030178) obtained using Whittaker's root series formula.

1, 1, -1, -5, 19, -3, -10187, 146847, 3268961, -211632497, 393324007, 5402916117, -3884618921299, -774402304798329, 148294948981707557, -3311395903665985169, -43463254022673425965, 14469962812566878696039, 6554498075974546253080309, -3074689522272735111427973673

Offset: 1

Author

Raul Prisacariu, Feb 19 2024

Keywords

• sign

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 numerators of the infinite series.

Examples

			a(1) is the numerator of -1/-2 = 1/2.
a(2) is the numerator of -(1/2)/((-2)*det ToeplitzMatrix((-2,1),(-2,1/2!))) = -(1/2)/((-2)*(7/2)) = 1/14.
a(3) is the numerator 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, A370490 (denominator).

Formula

For n > 1, a(n) is the numerator 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(20) from Chai Wah Wu, Mar 23 2024