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 A176599 #37 May 08 2023 09:36:24
%S A176599 1,-1,-1,-5,-7,-1631,-96547,-40291823,-16870575007,-7075000252463,
%T A176599 -2969301738826267,-13713149169712887583,-10557203537780702505907
%N A176599 Numerators of the first column of a table with top row the nonnegative integers and successive rows defined by a reverted Akiyama-Tanigawa procedure.
%C A176599 Define a reverted Akiyama-Tanigawa procedure which takes a sequence s(1), s(2), s(3), ..., as input and constructs the sequence of (s(k)-s(k+1))/k as output. (The difference from the standard algorithm is that the differences are divided by k, not multiplied by k.)
%C A176599 Starting from a top row with nonnegative integers, the following table is constructed row after row by applying the reverted algorithm in succession:
%C A176599 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, ...
%C A176599 -1, -1/2, -1/3, -1/4, -1/5, -1/6, -1/7, -1/8, -1/9, -1/10, -1/11, ...
%C A176599 -1/2, -1/12, -1/36, -1/80, -1/150, -1/252, -1/392, -1/576, -1/810, ...
%C A176599 -5/12, -1/36, -11/2160, -7/4800, -17/31500, -5/21168, -23/197568, ...
%C A176599 -7/18, -49/4320, -157/129600, -463/2016000, -803/13230000, ...
%C A176599 -1631/4320, -1313/259200, -17813/54432000, -35767/846720000, ...
%C A176599 -96547/259200, -257917/108864000, -2171917/22861440000, ...
%C A176599 The numerators of the left column define the current sequence.
%C A176599 The denominators of the third row are in A011379.
%H A176599 Cormac O'Sullivan, <a href="https://arxiv.org/abs/2208.02898">Stirling's approximation and a hidden link between two of Ramanujan's approximations</a>, arXiv:2208.02898 [math.NT], 2022. See p. 17.
%H A176599 Craig A. Tracy and H. Widom, <a href="http://arxiv.org/abs/1601.04677">On the ground state energy of the delta-function Bose gas</a>, arXiv preprint arXiv:1601.04677 [math-ph], 2016.
%F A176599 From _Peter Bala_, Aug 14 2012: (Start)
%F A176599 The o.g.f. for the rational numbers in the first column of the above table is Sum_{n >= 0} x^n/(Product_{k = 1..n} (x-k)) = 1 - x - 1/2*x^2 - 5/12*x^3 - 7/18*x^4 - .... This yields the formula |a(n)| = numerator of Sum_{k = 0..n-1} (1/k!) * Sum_{i = 0..k} (-1)^i*binomial(k,i)*(k-i+1)^(k-n). Cf. A024427.
%F A176599 More generally, the o.g.f. for the rational numbers in the r-th column of the above table (excluding the first entry of r) is Sum_{n >= r} x^(n+1-r)/ (Product_{k = r..n} (x-k)).
%F A176599 (End)
%F A176599 The first column of the above table lists the coefficients of the expansion of b(1)x/(1+b(2)x/(1+b(3)x/(1+b(4)x/(...)))), a continued fraction, where b(n) are -1, -1/2, -1/3, -1/4, ... i.e., the second row of the table above. - _Benedict W. J. Irwin_, May 10 2016
%e A176599 From _Peter Bala_, Aug 14 2012: (Start)
%e A176599 Column 2: Sum_{n >= 2} x^(n-1)/(Product_{k = 2..n} (x-k)) = -(1/2)*x - (1/12)*x^2 - (1/36)*x^3 - (49/4320)*x^4 - ...
%e A176599 Column 3: Sum_{n >= 3} x^(n-2)/(Product_{k = 3..n} (x-k)) = -(1/3)*x - (1/36)*x^2 - (11/2160)*x^3 - (157/129600)*x^4 - .... (End)
%t A176599 a[1, k_] := k; a[n_, k_] := a[n, k] = (a[n-1, k] - a[n-1, k+1])/k; a[n_] := Numerator[a[n, 1]]; Table[a[n], {n, 1, 13}] (* _Jean-François Alcover_, Aug 02 2012 *)
%Y A176599 Cf. A024427.
%K A176599 frac,sign
%O A176599 0,4
%A A176599 _Paul Curtz_, Apr 21 2010