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.
1, -1, -1, -5, -7, -1631, -96547, -40291823, -16870575007, -7075000252463, -2969301738826267, -13713149169712887583, -10557203537780702505907
Offset: 0
Examples
From _Peter Bala_, Aug 14 2012: (Start)
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 - ...
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)
Links
- Cormac O'Sullivan, Stirling's approximation and a hidden link between two of Ramanujan's approximations, arXiv:2208.02898 [math.NT], 2022. See p. 17.
- Craig A. Tracy and H. Widom, On the ground state energy of the delta-function Bose gas, arXiv preprint arXiv:1601.04677 [math-ph], 2016.
Crossrefs
Cf. A024427.
Programs
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Mathematica
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 *)
Formula
From Peter Bala, Aug 14 2012: (Start)
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.
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)).
(End)
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
Comments