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

0,4

Comments

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.)

Starting from a top row with nonnegative integers, the following table is constructed row after row by applying the reverted algorithm in succession:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, ...

-1, -1/2, -1/3, -1/4, -1/5, -1/6, -1/7, -1/8, -1/9, -1/10, -1/11, ...

-1/2, -1/12, -1/36, -1/80, -1/150, -1/252, -1/392, -1/576, -1/810, ...

-5/12, -1/36, -11/2160, -7/4800, -17/31500, -5/21168, -23/197568, ...

-7/18, -49/4320, -157/129600, -463/2016000, -803/13230000, ...

-1631/4320, -1313/259200, -17813/54432000, -35767/846720000, ...

-96547/259200, -257917/108864000, -2171917/22861440000, ...

The numerators of the left column define the current sequence.

The denominators of the third row are in A011379.

Links

Table of n, a(n) for n=0..12.

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.

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

Example

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)

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 *)

Crossrefs

Cf. A024427.

Sequence in context: A056252 A274774 A370190 * A309409 A291687 A257774

Adjacent sequences: A176596 A176597 A176598 * A176600 A176601 A176602

Keyword

frac,sign

Author

Paul Curtz, Apr 21 2010

Status

approved