A174727 -id:A174727 - cp's OEIS frontend

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

Paul Curtz, Aug 27 2011

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).

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.

Cf. A194506 (denominator).

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

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 *)

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

A232853 Repeat n+1 times A091137(n).

Original entry on oeis.org

1, 2, 2, 12, 12, 12, 24, 24, 24, 24, 720, 720, 720, 720, 720, 1440, 1440, 1440, 1440, 1440, 1440, 60480, 60480, 60480, 60480, 60480, 60480, 60480, 120960, 120960, 120960, 120960, 120960, 120960, 120960, 120960, 3628800

Offset: 0

Paul Curtz, Dec 01 2013

A002657(n) and A091137(n) are linked to the Bernoulli numbers B n.
Unreduced differences table of A002657(n)/A091137(n):
1,            1/2,     5/12,    9/24,  251/720, 475/1440,...
-1/2,       -1/12,    -1/24, -19/720, -27/1440,... =-A141417(n+1)/A091137(n+1),
5/12,        1/24,   11/720, 11/1440,...
-9/24,    -19/720, -11/1440,...
251/720,  27/1440,...
-475/1440,... etc.
This is an autosequence of the second kind: its inverse binomial transform is the signed sequence with the main diagonal double of the first upper diagonal.
a(n) is the denominators written by antidiagonals.

			1,
2,   2,
12, 12, 12,
24, 24, 24, 24, etc.

P. Curtz, Intégration numérique des systèmes différentiels à conditions initiales. Centre de Calcul Scientifique de l'Armement, Arcueil 1969. Pages 36 and 56.

Cf. A195287, A002208/A002209 (reduced autosequence), A193546, A174727, A165313.

Repeat n+1 times A091137(n). Triangle.

A176493 A091137(n)/(n+1).

Original entry on oeis.org

1, 1, 4, 6, 144, 240, 8640, 15120, 403200, 725760, 43545600, 79833600, 201180672000, 373621248000, 2092278988800, 3923023104000, 1883051089920000, 3556874280960000, 2688996956405760000, 5109094217170944000, 1605715325396582400000, 3065456530302566400000

Offset: 0

Paul Curtz, Apr 19 2010

nonn

a(n) = A174727(n)/A027760(n+1).

cp's OEIS Frontend

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

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A232853 Repeat n+1 times A091137(n).

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A176493 A091137(n)/(n+1).

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