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

1, 1, 7, 17, 41, 731, 8563, 27719, 190073, 516149, 1013143139, 1519024289, 14108351869, 14399405173, 23142912688967, 83945247395407, 84894728616107, 3204549982389941, 262488267575333123, 9027726081126601799, 2026692221793223022131, 1375035304877251309001

Offset: 0

Author

Paul Curtz, Aug 27 2011

Keywords

• nonn
• frac

Comments

Akiyama-Tanigawa from 1/n gives Bernoulli A164555(n)/A027642(n).

Reciprocally

1, 1/2, 5/12, 3/8, 251/720, 95/288, 19087/60480, 5257/17280,

1/2, 1/6, 1/8, 19/180, 3/32, 863/10080, 275/3456,

1/3, 1/12, 7/120, 17/360, 41/1008, 731/20160, 8563/259200,

1/4, 1/20, 1/30, 11/420, 89/4032,5849/302400,

1/5, 1/30, 3/140, 83/5040, 59/4320,

1/6, 1/42, 5/336,

1/7, 1/56,

1/8.

First row: A002208/A002209 or reduced A002657(n)/A091137(n) unsigned.

Second row: A002206(n+1)/A002689(n) unsigned. See A141417(n) and A174727(n).

Third row: a(n)/A194506(n).

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200
• Iaroslav V. Blagouchine, Three notes on Ser's and Hasse's representation for the zeta-functions, Integers (2018) 18A, Article #A3.

Crossrefs

Cf. A194506 (denominator).

Programs

Maple

read("transforms3") ;
L := [seq(1/n,n=1..20)] ;
L1 := AKIYAMATANIGAWAi(L) ;
L2 := AKIYATANI(L1) ;
L3 := AKIYATANI(L2) ;
apply(numer,%) ; # R. J. Mathar, Aug 27 2011
# second Maple program:
b:= proc (n, k) option remember;
      `if`(n=0, 1/(k+1), b(n-1, k) -b(n-1, k+1)/n)
    end:
a:= n-> numer(b(n, 2)):
seq(a(n), n=0..30);  # Alois P. Heinz, Aug 27 2011

Mathematica

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 *)
Numerator@Table[(-1)^n (n + 1) Integrate[FunctionExpand[x Binomial[x, n + 1]], {x, 0, 1}], {n, 0, 20}] (* Vladimir Reshetnikov, Feb 01 2017 *)

Formula

a(n)/A194506(n) = (-1)^n * (n+1) * Integral_{0Vladimir Reshetnikov, Feb 01 2017