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 A370490 #23 May 30 2025 03:32:29 %S A370490 2,14,259,9657,200187,18671081,7313976065,1273374259615, %T A370490 285038137030769,79755360301275363,9091712937155442435, %U A370490 149243024021521700285,1085736156475373087072485,3071709182054627484879798019,2005459027715242401528647218817,1496371535371115486607560677791759 %N A370490 The denominators of a series that converges to the Omega constant (A030178) obtained using Whittaker's root series formula. %C A370490 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. %H A370490 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 A370490 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!. %e A370490 a(1) is the denominator of -1/-2 = 1/2. %e A370490 a(2) is the denominator of -(1/2)/((-2)*det ToeplitzMatrix((-2,1),(-2,1/2!))) = -(1/2)/((-2)*(7/2)) = 1/14. %e A370490 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. %Y A370490 Cf. A030178, A370491 (numerator). %K A370490 nonn %O A370490 1,1 %A A370490 _Raul Prisacariu_, Feb 19 2024 %E A370490 a(9)-a(16) from _Chai Wah Wu_, Mar 23 2024