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%I #26 Apr 06 2016 09:15:51
%S -1,0,3,4,5,1,7,1,1,1,11,1,13,1,1,1,17,1,19,1,1,1,23,1,1,1,1,1,29,1,
%T 31,1,1,1,1,1,37,1,1,1,41,1,43,1,1,1,47,1,1,1,1,1,53,1,1,1,1,1,59,1,
%U 61,1,1,1,1,1,67,1,1,1,71,1,73,1,1,1,1,1,79
%N Numerators of the second row of the Akiyama-Tanigawa table for the sequence 1/n!.
%C Filling the top row of a table with T(0,k) = 1/k!, k>=0, the Akiyama-Tanigawa algorithm constructs the following table T(n,k) of fractions, n>=0, k>=0:
%C 1, 1, 1/2, 1/6, 1/24, 1/120, 1/720, 1/5040, 1/40320, 1/362880,...
%C 0, 1, 1, 1/2, 1/6, 1/24, 1/120, 1/720, 1/5040, 1/40320, 1/362880, ...
%C -1, 0, 3/2, 4/3, 5/8, 1/5, 7/144, 1/105, 1/640, 1/4536, 11/403200, ...
%C -1, -3, 1/2, 17/6, 17/8, 109/120, 197/720, 107/1680, 487/40320, ..
%C 2, -7, -7, 17/6, 73/12, 457/120, 529/360, 2081/5040, 263/2880,...
%C 9, 0, -59/2, -13, 91/8, 421/30, 355/48, 2161/840, 3871/5760, 709/5040, ..
%C 9, 59, -99/2, -195/2, -319/24, 1593/40, 2701/80, 76631/5040, 4285/896,...
%C The numerators of T(2,k) are the current sequence.
%C The denominators are 1, 1, 2, 3, 8, 5, 144, 105, 640, 4536, 403200, 332640, 43545600, 37065600,...
%C T(0,k) = T(1,k+1), shifted.
%C The left column is T(n,0) = (-1)^(n+1)*A014182(n).
%C The column T(n,1) appears to be (-1)^n*A074051(n). - _R. J. Mathar_, Jan 16 2011
%C a(n) = numerator(A005563(n-1)/(n-1)!), for n>0. - _Fred Daniel Kline_, Mar 20 2016
%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.
%t nn = 78; Numerator[Simplify[CoefficientList[Series[-Zeta[x] + (Derivative[1][Zeta][x] + x*Derivative[2][Zeta][x])*x, {x, 0, nn}], x]/Table[Derivative[n][Zeta][0], {n, 0, nn}]]] (* _Mats Granvik_, Nov 11 2013 *)
%Y Cf. A089026, A090585, A080305.
%K frac,sign
%O 0,3
%A _Paul Curtz_, Mar 21 2010
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