This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A370491 #22 May 30 2025 03:32:34 %S A370491 1,1,-1,-5,19,-3,-10187,146847,3268961,-211632497,393324007, %T A370491 5402916117,-3884618921299,-774402304798329,148294948981707557, %U A370491 -3311395903665985169,-43463254022673425965,14469962812566878696039,6554498075974546253080309,-3074689522272735111427973673 %N A370491 The numerators of a series that converges to the Omega constant (A030178) obtained using Whittaker's root series formula. %C A370491 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 A370491 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 A370491 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 A370491 a(1) is the numerator of -1/-2 = 1/2. %e A370491 a(2) is the numerator of -(1/2)/((-2)*det ToeplitzMatrix((-2,1),(-2,1/2!))) = -(1/2)/((-2)*(7/2)) = 1/14. %e A370491 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 A370491 Cf. A030178, A370490 (denominator). %K A370491 sign %O A370491 1,4 %A A370491 _Raul Prisacariu_, Feb 19 2024 %E A370491 a(9)-a(20) from _Chai Wah Wu_, Mar 23 2024