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A174129 Numerators of the first column of the table of fractions generated by the Akiyama-Tanigawa transform from a first row A164555(k)/A027642(k).

%I A174129 #28 Feb 18 2022 23:09:01
%S A174129 1,1,-1,-1,31,7,-1051,-201,56911,18311,-24346415,-4227881,
%T A174129 425739604981,2082738855,-759610463437,-1935668684041,
%U A174129 91825384886337407,3104887811293639,-333936446105326262497,-8039608511660213481,496858217433153341005061
%N A174129 Numerators of the first column of the table of fractions generated by the Akiyama-Tanigawa transform from a first row A164555(k)/A027642(k).
%C A174129 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:
%C A174129   1, 1/2, 1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0, 5/66, 0, -691/2730, 0, 7/6, ...
%C A174129   1/2, 2/3, 1/2, 2/15, -1/6, -1/7, 1/6, 4/15, -3/10, -25/33, 5/6, 1382/455, ...
%C A174129   -1/6, 1/3, 11/10, 6/5, -5/42, -13/7, -7/10, 68/15, 453/110, -175/11, ...
%C A174129   -1/2, -23/15, -3/10, 554/105, 365/42, -243/35, -1099/30, 548/165, 19827/110, ...
%C A174129   31/30, -37/15, -1171/70, -478/35, 469/6, 1247/7, -6153/22, -46708/33, ...
%C A174129   7/2, 599/21, -129/14, -38566/105, -20995/42, 211515/77, 524699/66, ...
%C A174129 The numerators of the leftmost column define the current sequence.
%H A174129 D. Merlini, R. Sprugnoli and M. C. Verri, <a href="https://www.emis.de/journals/INTEGERS/papers/f5/f5.Abstract.html">The Akiyama-Tanigawa Transformation</a>, Integers, 5 (1) (2005) #A05.
%F A174129 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
%p A174129 read("transforms3") ;
%p A174129 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:
%t A174129 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 *)
%Y A174129 Cf. A141056 (denominators), A174110, A174111 (first row).
%Y A174129 Cf. A164555, A027642, A048993.
%K A174129 frac,sign
%O A174129 0,5
%A A174129 _Paul Curtz_, Mar 09 2010