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%I #37 Sep 04 2018 12:42:58
%S 1,1,7,17,41,731,8563,27719,190073,516149,1013143139,1519024289,
%T 14108351869,14399405173,23142912688967,83945247395407,84894728616107,
%U 3204549982389941,262488267575333123,9027726081126601799,2026692221793223022131,1375035304877251309001
%N Numerator of the third row of the inverse Akiyama-Tanigawa algorithm from 1/n.
%C Akiyama-Tanigawa from 1/n gives Bernoulli A164555(n)/A027642(n).
%C Reciprocally
%C 1, 1/2, 5/12, 3/8, 251/720, 95/288, 19087/60480, 5257/17280,
%C 1/2, 1/6, 1/8, 19/180, 3/32, 863/10080, 275/3456,
%C 1/3, 1/12, 7/120, 17/360, 41/1008, 731/20160, 8563/259200,
%C 1/4, 1/20, 1/30, 11/420, 89/4032,5849/302400,
%C 1/5, 1/30, 3/140, 83/5040, 59/4320,
%C 1/6, 1/42, 5/336,
%C 1/7, 1/56,
%C 1/8.
%C First row: A002208/A002209 or reduced A002657(n)/A091137(n) unsigned.
%C Second row: A002206(n+1)/A002689(n) unsigned. See A141417(n) and A174727(n).
%C Third row: a(n)/A194506(n).
%H Alois P. Heinz, <a href="/A193546/b193546.txt">Table of n, a(n) for n = 0..200</a>
%H Iaroslav V. Blagouchine, <a href="http://math.colgate.edu/~integers/sjs3/sjs3.Abstract.html">Three notes on Ser's and Hasse's representation for the zeta-functions</a>, Integers (2018) 18A, Article #A3.
%F a(n)/A194506(n) = (-1)^n * (n+1) * Integral_{0<x<1} x*binomial(x,n+1). - _Vladimir Reshetnikov_, Feb 01 2017
%p read("transforms3") ;
%p L := [seq(1/n,n=1..20)] ;
%p L1 := AKIYAMATANIGAWAi(L) ;
%p L2 := AKIYATANI(L1) ;
%p L3 := AKIYATANI(L2) ;
%p apply(numer,%) ; # _R. J. Mathar_, Aug 27 2011
%p # second Maple program:
%p b:= proc (n, k) option remember;
%p `if`(n=0, 1/(k+1), b(n-1, k) -b(n-1, k+1)/n)
%p end:
%p a:= n-> numer(b(n, 2)):
%p seq(a(n), n=0..30); # _Alois P. Heinz_, Aug 27 2011
%t a[n_, 0] := 1/(n+1); a[n_, m_] := a[n, m] = a[n, m-1] - a[n+1, m-1]/m; Table[a[2, m], {m, 0, 21}] // Numerator (* _Jean-François Alcover_, Aug 09 2012 *)
%t Numerator@Table[(-1)^n (n + 1) Integrate[FunctionExpand[x Binomial[x, n + 1]], {x, 0, 1}], {n, 0, 20}] (* _Vladimir Reshetnikov_, Feb 01 2017 *)
%Y Cf. A194506 (denominator).
%K nonn,frac
%O 0,3
%A _Paul Curtz_, Aug 27 2011
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