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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. 0
1, -1, -1, -5, -7, -1631, -96547, -40291823, -16870575007, -7075000252463, -2969301738826267, -13713149169712887583, -10557203537780702505907 (list; graph; refs; listen; history; text; internal format)
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 through 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.

Craig A. Tracy, H Widom, On the ground state energy of the delta-function Bose gas, arXiv preprint arXiv:1601.04677, 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 are 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

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 - .... - Peter Bala, Aug 14 2012

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-Fran├žois Alcover, Aug 02 2012 *)

CROSSREFS

Cf. A024427.

Sequence in context: A101829 A056252 A274774 * A309409 A291687 A257774

Adjacent sequences:  A176596 A176597 A176598 * A176600 A176601 A176602

KEYWORD

frac,sign

AUTHOR

Paul Curtz, Apr 21 2010

STATUS

approved

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Last modified April 15 04:53 EDT 2021. Contains 342975 sequences. (Running on oeis4.)