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A193546 Numerator of the third row of the inverse Akiyama-Tanigawa algorithm from 1/n.

%I A193546 #37 Sep 04 2018 12:42:58
%S A193546 1,1,7,17,41,731,8563,27719,190073,516149,1013143139,1519024289,
%T A193546 14108351869,14399405173,23142912688967,83945247395407,84894728616107,
%U A193546 3204549982389941,262488267575333123,9027726081126601799,2026692221793223022131,1375035304877251309001
%N A193546 Numerator of the third row of the inverse Akiyama-Tanigawa algorithm from 1/n.
%C A193546 Akiyama-Tanigawa from 1/n gives Bernoulli A164555(n)/A027642(n).
%C A193546 Reciprocally
%C A193546 1,   1/2,   5/12,     3/8, 251/720,    95/288, 19087/60480, 5257/17280,
%C A193546 1/2, 1/6,    1/8,  19/180,    3/32, 863/10080,    275/3456,
%C A193546 1/3, 1/12, 7/120,  17/360, 41/1008, 731/20160, 8563/259200,
%C A193546 1/4, 1/20, 1/30,   11/420, 89/4032,5849/302400,
%C A193546 1/5, 1/30, 3/140, 83/5040, 59/4320,
%C A193546 1/6, 1/42, 5/336,
%C A193546 1/7, 1/56,
%C A193546 1/8.
%C A193546 First row: A002208/A002209 or reduced A002657(n)/A091137(n) unsigned.
%C A193546 Second row: A002206(n+1)/A002689(n) unsigned. See A141417(n) and A174727(n).
%C A193546 Third row: a(n)/A194506(n).
%H A193546 Alois P. Heinz, <a href="/A193546/b193546.txt">Table of n, a(n) for n = 0..200</a>
%H A193546 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 A193546 a(n)/A194506(n) = (-1)^n * (n+1) * Integral_{0<x<1} x*binomial(x,n+1). - _Vladimir Reshetnikov_, Feb 01 2017
%p A193546 read("transforms3") ;
%p A193546 L := [seq(1/n,n=1..20)] ;
%p A193546 L1 := AKIYAMATANIGAWAi(L) ;
%p A193546 L2 := AKIYATANI(L1) ;
%p A193546 L3 := AKIYATANI(L2) ;
%p A193546 apply(numer,%) ; # _R. J. Mathar_, Aug 27 2011
%p A193546 # second Maple program:
%p A193546 b:= proc (n, k) option remember;
%p A193546       `if`(n=0, 1/(k+1), b(n-1, k) -b(n-1, k+1)/n)
%p A193546     end:
%p A193546 a:= n-> numer(b(n, 2)):
%p A193546 seq(a(n), n=0..30);  # _Alois P. Heinz_, Aug 27 2011
%t A193546 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 A193546 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 A193546 Cf. A194506 (denominator).
%K A193546 nonn,frac
%O A193546 0,3
%A A193546 _Paul Curtz_, Aug 27 2011