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A193546 Numerator of the third row of the inverse Akiyama-Tanigawa algorithm from 1/n. 4
1, 1, 7, 17, 41, 731, 8563, 27719, 190073, 516149, 1013143139, 1519024289, 14108351869, 14399405173, 23142912688967, 83945247395407, 84894728616107, 3204549982389941, 262488267575333123, 9027726081126601799, 2026692221793223022131, 1375035304877251309001 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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.

FORMULA

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

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-Fran├žois 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 *)

CROSSREFS

Cf. A194506 (denominator).

Sequence in context: A193214 A184862 A194772 * A268255 A124965 A253973

Adjacent sequences:  A193543 A193544 A193545 * A193547 A193548 A193549

KEYWORD

nonn,frac

AUTHOR

Paul Curtz, Aug 27 2011

STATUS

approved

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Last modified June 20 20:22 EDT 2021. Contains 345233 sequences. (Running on oeis4.)