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%I #19 Mar 24 2024 02:25:50
%S 1,1,-1,-5,19,-3,-10187,146847,3268961,-211632497,393324007,
%T 5402916117,-3884618921299,-774402304798329,148294948981707557,
%U -3311395903665985169,-43463254022673425965,14469962812566878696039,6554498075974546253080309,-3074689522272735111427973673
%N The numerators of a series that converges to the Omega constant (A030178) obtained using Whittaker's root series formula.
%C 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.
%H E. T. Whittaker and G. Robinson, <a href="https://archive.org/details/calculusofobserv031400mbp/page/n139/mode/2up">The Calculus of Observations</a>, London: Blackie & Son, Ltd. 1924, pp. 120-123.
%F 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!.
%e a(1) is the numerator of -1/-2 = 1/2.
%e a(2) is the numerator of -(1/2)/((-2)*det ToeplitzMatrix((-2,1),(-2,1/2!)) = -(1/2)/((-2)(7/2)) = 1/14.
%e 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.
%Y Cf. A030178, A370490 (denominator).
%K sign
%O 1,4
%A _Raul Prisacariu_, Feb 19 2024
%E a(9)-a(20) from _Chai Wah Wu_, Mar 23 2024