login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

Numerators of the second row of the Akiyama-Tanigawa table for the sequence 1/n!.
1

%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