A193220 -id:A193220 - cp's OEIS frontend

A194531 Numerator of row 4 in A051714(n) or row 3 in A176672(n).

0, 1, 1, 2, 5, 5, 7, 28, 3, 15, 55, 22, 13, 91, 35, 40, 34, 51, 57, 190, 35, 77, 253, 92, 25, 325, 117, 126, 203, 145, 155, 496, 44, 187, 595, 210, 111, 703, 247, 260, 205, 287, 301, 946, 165, 345, 1081, 376, 98, 1225, 425

Offset: 0

Paul Curtz, Aug 28 2011

Akiyama-Tanigawa algorithm from 1/n leads to Bernoulli A164555(n)/A027642(n):
1,   1/2,  1/3,  1/4,
1/2, 1/3,  1/4,  1/5,
1/6, 1/6, 3/20, 2/15,           =A026741(n+1)/A045896(n+1),
0,  1/30, 1/20, 2/35, 5/84, 5/84, 7/120, 28/495  =a(n)/b(n).

Cf. A193220 (denominators).

a[0, k_] := 1/(k+1); a[n_, k_] := a[n, k] = (k+1)*(a[n-1, k] - a[n-1, k+1]); Table[a[3, k], {k, 0, 50}] // Numerator (* Jean-François Alcover, Sep 19 2012 *)

A100652 Denominator of 1 - Sum_{i=1..n} |Bernoulli(i)|.

Original entry on oeis.org

1, 2, 3, 3, 10, 10, 105, 105, 70, 70, 1155, 1155, 1430, 1430, 2145, 2145, 24310, 24310, 4849845, 4849845, 58786, 58786, 2028117, 2028117, 965770, 965770, 1448655, 1448655, 28007330, 28007330, 100280245065, 100280245065, 66853496710, 66853496710, 100280245065

Offset: 1

N. J. A. Sloane, Dec 05 2004

Contribution from Paul Curtz, Aug 07 2012 (Start):
Take a(0)=1. Then instead of the Akiyama-Tanigawa algorithm we create the extended (or prolonged) Akiyama-Tanigawa algorithm using A028310(n)=1,1,2,3,4,5,... instead of A000027(n)=1,2,3,4,5,.. .
Hence the array (A051714 with an additional column)
2,       1,   1/2,   1/3,     1/4,
1,     1/2,   1/3,   1/4,     1/5,
1/2,   1/6,   1/6,   3/20,   2/15,       A026741(n+1)/A045896(n+1)
1/3,     0,  1/30,   1/20,   2/35,       A194531(n)/A193220(n)
1/3, -1/30, -1/30, -3/140, -1/105.       A051722(n)/A051723(n).
a(n) is the denominator of the (first) column before the Akiyama-Tanigawa algorithm leading to the second Bernoulli numbers A164555(n)/A027642(n). See A176672(n).
(End)

			1, 1/2, 1/3, 1/3, 3/10, 3/10, 29/105, 29/105, 17/70, 17/70, 193/1155, 193/1155, -123/1430, -123/1430, -2687/2145, -2687/2145, -202863/24310, -202863/24310, -307072861/4849845, ... = A100651/A100652.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Denominator[1-(Accumulate[Abs[BernoulliB[Range[0,40]]]])] (* Harvey P. Dale, Jan 28 2013 *)

A230262 Numerators of Akiyama-Tanigawa algorithm applied to harmonic numbers, written by antidiagonals.

Original entry on oeis.org

1, 3, -1, 11, -2, 1, 25, -3, 1, 0, 137, -4, 3, 1, -1, 49, -5, 2, 1, -1, 0, 363, -6, 5, 2, -3, -1, 1, 761, -7, 3, 5, -1, -1, 1, 0, 7129, -8, 7, 5, 0, -4, 1, 1, -1, 7381, -9, 4, 7, 1, -1, -1, 1, -1, 0, 83711, -10, 9, 28, 49, -29, -5, 8, 1, -5, 5

Offset: 0

Paul Curtz, Nov 09 2013

Leading column gives the Bernoulli numbers: A027641(n)/A027642(n). In A051714, A164555 must be written instead of A027641.

			Numerators of
1,    3/2, 11/6, 25/12,...
-1/2, -2/3, -3/4,  -4/5,...
1/6,   1/6, 3/20,  2/15,... =A026741(n+1)/A045896(n+1)
0,    1/30, 1/20,  2/35,... =A194531/A193220.

M. Kaneko, The Akiyama-Tanigawa algorithm for Bernoulli numbers, J. Integer Sequences, Vol 3 (2000), #00.2.9.

Cf. A051714/A051715, A001008/A002805.

t[1, k_] := HarmonicNumber[k]; t[n_, k_] := t[n, k] = k*(t[n-1, k] - t[n-1, k+1]); Table[t[n-k+1, k] // Numerator, {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Nov 15 2013 *)

More terms from Jean-François Alcover, Nov 15 2013

cp's OEIS Frontend

A194531 Numerator of row 4 in A051714(n) or row 3 in A176672(n).

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A100652 Denominator of 1 - Sum_{i=1..n} |Bernoulli(i)|.

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A230262 Numerators of Akiyama-Tanigawa algorithm applied to harmonic numbers, written by antidiagonals.

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