
COMMENTS

Define a reverted AkiyamaTanigawa 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.


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} (xk)) = 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..n1} 1/k!*sum {i = 0..k} (1)^i*binomial(k,i)*(ki+1)^(kn). Cf. A024427.
More generally, the o.g.f. for the rational numbers in the rth column of the above table (excluding the first entry of r) is sum {n >= r} x^(n+1r)/(product {k = r..n} (xk)).
(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
