OFFSET
0,5
COMMENTS
The first 6 rows if the table generated by iterative application of the Akiyama-Tanigawa transform starting with a header row of fractions A164555(k)/A027642(k) are:
1, 1/2, 1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0, 5/66, 0, -691/2730, 0, 7/6, ...
1/2, 2/3, 1/2, 2/15, -1/6, -1/7, 1/6, 4/15, -3/10, -25/33, 5/6, 1382/455, ...
-1/6, 1/3, 11/10, 6/5, -5/42, -13/7, -7/10, 68/15, 453/110, -175/11, ...
-1/2, -23/15, -3/10, 554/105, 365/42, -243/35, -1099/30, 548/165, 19827/110, ...
31/30, -37/15, -1171/70, -478/35, 469/6, 1247/7, -6153/22, -46708/33, ...
7/2, 599/21, -129/14, -38566/105, -20995/42, 211515/77, 524699/66, ...
The numerators of the leftmost column define the current sequence.
LINKS
D. Merlini, R. Sprugnoli and M. C. Verri, The Akiyama-Tanigawa Transformation, Integers, 5 (1) (2005) #A05.
FORMULA
a(n) = numerator(Sum_{j=0..n} (-1)^(n-j)*j!*Stirling2(n,j)*B(j)), where B are the Bernoulli numbers A164555/A027642. - Fabián Pereyra, Jan 06 2022
MAPLE
read("transforms3") ;
A174129 := proc(n) Lin := [bernoulli(0), -bernoulli(1), seq(bernoulli(k), k=2..n+1)] ; for r from 1 to n do Lin := AKIYATANI(Lin) ; end do; numer(op(1, Lin)) ; end proc:
MATHEMATICA
a[0, k_] := a[0, k] = BernoulliB[k]; a[0, 1] = 1/2; a[n_, k_] := a[n, k] = (k+1)*(a[n-1, k] - a[n-1, k+1]); a[n_] := a[n, 0] // Numerator; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Aug 14 2012 *)
CROSSREFS
KEYWORD
frac,sign
AUTHOR
Paul Curtz, Mar 09 2010
STATUS
approved