%I #37 Feb 27 2022 15:48:55
%S 1,1,13,7,149,157,383,199,7409,7633,86231,88331,1173713,1197473,
%T 1219781,620401,42862943,43503583,279379879,283055551,57313183,
%U 19328341,449489867,1362695813,34409471059,34738962067,315510823603,45467560829,9307359944587,9382319148907,293103346860157,147643434162641,594812856101039,54448301591149
%N Numerators of the Inverse Akiyama-Tanigawa transform of the aerated even-indexed Bernoulli numbers 1, 0, 1/6, 0, -1/30, 0, 1/42, ...
%C These are the numerators of the first row of a Table T(n,k) which contains the even-indexed Bernoulli numbers in the first column: T(2n,0) = A000367(n)/A002445(n), T(2n+1,0)=0, and which generates rows with the Akiyama-Tanigawa transform. (Because the first column is given, the algorithm is an inverse Akiyama-Tanigawa transform.)
%C These are the absolute values of the numerators of the Taylor expansion of sinh(log(x+1))*log(x+1)at x=0. - _Gary Detlefs_, Aug 31 2011
%H L. A. Medina, V. H. Moll, E. S. Rowland, <a href="http://arxiv.org/abs/0911.1325">Iterated primitives of logarithmic powers</a>, arXiv:0911.1325, arXiv:0911.1325 [math.NT], 2009-2010.
%H D. Merlini, R. Sprugnoli, M. C. Verri, <a href="http://www.emis.de/journals/INTEGERS/papers/f5/f5.Abstract.html">The Akiyama-Tanigawa Transformation</a>, Integers, 5 (1) (2005) #A05.
%F From _Groux Roland_, Jan 07 2011: (Start)
%F T(0,k) = H(k)/2 + 1/(k+1) with H(k) harmonic number of order k.
%F T(0,k)= -(1/2)*(k+1)*Integral_{x=0..1} x^n*log(x*(1-x)) dx.
%F G.f.: Sum_{k>=0} T(0,k) x^k = (x-2)*(log(1-x))/(2*x*(1-x)). (End)
%F T(1,n) = -A191567(n)/A061038(n+2) = -A060819(n)/A145979(n). - _Paul Curtz_, Jul 19 2011
%F (T(1,n))^2 = A181318(n)/A061038(n+2). - _Paul Curtz_, Jul 19 2011, index corrected by _R. J. Mathar_, Sep 09 2011
%e The table T(n,k) of fractions generated by the Akiyama-Tanigawa transform, with the column T(n,0) equal to Bernoulli(n) for even n and equal to 0 for odd n, starts in row n=0 as:
%e 1, 1, 13/12, 7/6, 149/120, 157/120, 383/280, 199/140, ...
%e 0, -1/6, -1/4, -3/10, -1/3, -5/14, -3/8, -7/18, -2/5, -9/22, ...
%e 1/6, 1/6, 3/20, 2/15, 5/42, 3/28, 7/72, 4/45, 9/110, 5/66, ...
%e 0, 1/30, 1/20, 2/35, 5/84, 5/84, 7/120, 28/495, 3/55, 15/286, ...
%e -1/30, -1/30, -3/140, -1/105, 0, 1/140, 49/3960, 8/495, ...
%e 0, -1/42, -1/28, -4/105, -1/28, -29/924, -7/264, -28/1287, -87/5005, ...
%e 1/42, 1/42, 1/140, -1/105, -5/231, -9/308, -343/10296, -1576/45045, ...
%t t[n_, 0] := BernoulliB[n]; t[1, 0]=0; t[n_, k_] := t[n, k] = (t[n, k-1] + (k-1)*t[n, k-1] - t[n+1, k-1])/k; Table[t[0, k], {k, 0, 33}] // Numerator (* _Jean-François Alcover_, Aug 09 2012 *)
%Y Cf. A177690 (denominators).
%K nonn,frac
%O 0,3
%A _Paul Curtz_, May 07 2010