A164555 -id:A164555 - cp's OEIS frontend

A181722 Numerator of (1/n - Bernoulli number A164555(n)/A027642(n)).

0, 0, 1, 1, 7, 1, 5, 1, 13, 1, 1, 1, 901, 1, -11, 1, 3647, 1, -43825, 1, 1222387, 1, -854507, 1, 1181821001, 1, -76977925, 1, 23749461059, 1, -8615841275543, 1, 28267510484519, 1

Offset: 1

Paul Curtz, Nov 17 2010

An autosequence is a sequence whose inverse binomial transform is the sequence signed. In integers, the oldest example is Fibonacci A000045. In fractions, A164555/A027642 is the son of 1/n via the Akiyama-Tanigawa algorithm; grandson is (A174110/A174111) = 1/2, 2/3, 1/2, 2/15, ...; see A164020. See A174341/A174342. All are from the same family.

			Fractions are 0, 0, 1/6, 1/4, 7/30, 1/6, 5/42, 1/8, 13/90, 1/10, 1/66, 1/12, 901/2730, ...

G. C. Greubel, Table of n, a(n) for n = 1..625
OEIS Wiki, Autosequence.

Cf. A000045, A027642, A164020, A164555, A174110, A174111, A174341, A174342.

A181722:= func< n | n le 2 select 0 else Numerator(1/n - Bernoulli(n-1)) >;
[A181722(n): n in [1..40]]; // G. C. Greubel, Mar 25 2024

a[n_] := If[n <= 2, 0, Numerator[1/n - BernoulliB[n-1]]];
Table[a[n], {n, 1, 34}] (* Jean-François Alcover, Jun 07 2017 *)

def A181722(n): return 0 if n<3 else numerator(1/n - bernoulli(n-1))
[A181722(n) for n in range(1,41)] # G. C. Greubel, Mar 25 2024

A165226 Numerator of 1 - A164555(n)/A027642(n).

Original entry on oeis.org

0, 1, 5, 1, 31, 1, 41, 1, 31, 1, 61, 1, 3421, 1, -1, 1, 4127, 1, -43069, 1, 174941, 1, -854375, 1, 236366821, 1, -8553097, 1, 23749461899, 1, -8615841261683, 1, 7709321041727, 1, -2577687858361, 1, 26315271553055396563, 1, -2929993913841553, 1

Offset: 0

Paul Curtz, Sep 09 2009

If n != 1, also the numerator of 1 - Bernoulli(n). The denominators are in A027642.
(There are no common factors to be canceled in the fractions.)
The numerators of 1 - Bernoulli(n) start 0, 3, 5,1, 31, ... and differ only at n=1 from this sequence.
E.g.f. for the rationals r(n) = a(n)/A027642(n) = 1 - A164555(n)/A027642(n): exp(x)*(1 - x/(exp(x) - 1)). - Wolfdieter Lang, Aug 07 2017

			The rationals r(n) begin: 0, 1/2, 5/6, 1, 31/30, 1, 41/42, 1, 31/30, 1, 61/66, 1, 3421/2730, 1, -1/6, 1, 4127/510, 1, -43069/798, 1, ... - _Wolfdieter Lang_, Aug 07 2017

Cf. A162173, A164555, A027642.

A165226 := proc(n) if n = 1 then 1+bernoulli(n) ; else 1-bernoulli(n) ; end if; numer(%) ; end proc: # R. J. Mathar, Jan 16 2011

|a(2n)| = A162173(n+1).

a(2n+1) = 1.

A172298 a(n) = A027641(n) * A164555(n).

Original entry on oeis.org

1, -1, 1, 0, 1, 0, 1, 0, 1, 0, 25, 0, 477481, 0, 49, 0, 13082689, 0, 1924313689, 0, 30489001321, 0, 730192467169, 0, 55867983514256281, 0, 73155570928609, 0, 564036899167989738841, 0, 74232720893311466588760025, 0, 59433630916551169012841089, 0, 6644474695172651051906689

Offset: 0

Paul Curtz, Jan 31 2010

sign

Squares of Bernoulli number numerators (apart from the sign flipped at a(1)).

The associated denominators of the squared Bernoulli numbers are in A172282.

Cf. A027641, A164555, A172282.

Edited and extended by R. J. Mathar, Feb 02 2010

A174342 Denominator of ( A164555(n)/A027642(n) + 1/(n+1) ).

Original entry on oeis.org

1, 1, 2, 4, 6, 6, 6, 8, 90, 10, 6, 12, 210, 14, 30, 16, 30, 18, 42, 20, 770, 22, 6, 24, 13650, 26, 54, 28, 30, 30, 462, 32, 5610, 34, 210, 36, 51870, 38, 26, 40, 330, 42, 42, 44, 2070, 46, 6, 48, 324870, 50, 1122, 52, 30, 54, 43890, 56, 5510, 58, 6, 60, 930930

Offset: 0

Paul Curtz, Mar 16 2010

The sequence A174341(n)/a(n) = 2, 1, 1/2, 1/4, 1/6, 1/6, 1/6, ... becomes 2, -1, 1/2, -1/4, 1/6,.. under inverse binomial transform: an autosequence, where each second term flips the sign.

OEIS Wiki, Autosequence

Cf. A174341 (numerators).

B(n)=if(n!=1, bernfrac(n), -bernfrac(n));
a(n)=denominator(B(n) + 1/(n + 1));
for(n=0, 60, print1(a(n),", ")) \\ Indranil Ghosh, Jun 19 2017

from sympy import bernoulli, Rational
def B(n):
    return bernoulli(n) if n != 1 else -bernoulli(n)
def a(n):
    return (B(n) + Rational(1, n + 1)).as_numer_denom()[1]
[a(n) for n in range(61)] # Indranil Ghosh, Jun 19 2017

A176144 a(2n) = A164555(n). a(2n+1) = A027641(n).

Original entry on oeis.org

1, 1, 1, -1, 1, 1, 0, 0, -1, -1, 0, 0, 1, 1, 0, 0, -1, -1, 0, 0, 5, 5, 0, 0, -691, -691, 0, 0, 7, 7, 0, 0, -3617, -3617, 0, 0, 43867, 43867, 0, 0, -174611, -174611, 0, 0, 854513, 854513, 0, 0, -236364091, -236364091, 0, 0, 8553103, 8553103, 0, 0, -23749461029, -23749461029, 0, 0, 8615841276005

Offset: 0

Paul Curtz, Apr 10 2010

Formally, these are the numerators of a sequence of fractions defined by alternating A164555(n)/A027642(n) with A027641(n)/A027642(n),
which apart from the third term duplicates the Bernoulli numbers.
Essentially a duplication of the entries of A027641.

Cf. A141056, A164020, A141056.

Edited by R. J. Mathar, Jun 07 2010

A193220 Denominators of the fourth row of Akiyama-Tanigawa algorithm leading to Bernoulli numbers A164555(n)/A027642(n).

Original entry on oeis.org

1, 30, 20, 35, 84, 84, 120, 495, 55, 286, 1092, 455, 280, 2040, 816, 969, 855, 1330, 1540, 5313, 1012, 2300, 7800, 2925, 819, 10962, 4060, 4495, 7440, 5456, 5984, 19635, 1785, 7770, 25308, 9139, 4940

Offset: 0

Paul Curtz, Jul 18 2011

Denominators of row k=3 of the table in A051714.

			The third row is 0, 1/30, 1/20, 2/35, 5/84, 5/84, 7/120, 28/495, 3/55, 15/286, 55/1092, 22/455, 13/280, ...

M. Kaneko, The Akiyama-Tanigawa algorithm for Bernoulli numbers, J. Integer Sequences, 3 (2000), #00.2.9.
D. Merlini, R. Sprugnoli, M. C. Verri, The Akiyama-Tanigawa Transformation, Integers, 5 (1) (2005) #A05.

Cf. A194531 (numerators).

read("transforms3");
L := [seq(1/n,n=1..40)] ;
L1 := AKIYATANI(L) ; L2 := AKIYATANI(L1) ; L3 := AKIYATANI(L2) ;
apply(denom,%) ; # R. J. Mathar, Aug 20 2011

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, 36}] // Denominator (* Jean-François Alcover, Sep 18 2012 *)

A235774 Let b(k) = A164555(k)/A027642(k), the sequence of "original" Bernoulli numbers with -1 instead of A164555(0)=1; then a(n) = numerator of the n-th term of the binomial transform of the b(k) sequence.

Original entry on oeis.org

-1, -1, 1, 1, 59, 3, 169, 5, 179, 7, 533, 9, 26609, 11, 79, 13, 3523, 15, 56635, 17, -168671, 19, 857273, 21, -236304031, 23, 8553247, 25, -23749438409, 27, 8615841677021, 29, -7709321025917, 31, 2577687858559, 33, -26315271552988224913

Offset: 0

Paul Curtz, Jan 15 2014

(a(n)/A027642(n)) = -1, -1/2, 1/6, 1, 59/30, 3, 169/42, 5, 179/30, 7, 533/66, 9,.. .
Difference table for a(n)/A027642(n):
-1, -1/2,   1/6,      1,  59/30,     3, 169/42, ...
1/2, 2/3,   5/6,  29/30,  31/30, 43/42,  41/42, ... = A165161(n)/A051717(n+1)
1/6, 1/6,  2/15,   1/15, -1/105, -1/21, -1/105, ... not in the OEIS
0, -1/30, -1/15, -8/105, -4/105, 4/105,  8/105, ... etc.
Compare with the array in A190339.

Cf. A164558/A027642, A165161(n)/A051717(n+1).

b[0] = -1; b[1] = 1/2; b[n_] := BernoulliB[n]; a[n_] := Sum[Binomial[n, k]*b[k], {k, 0, n}] // Numerator; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Jan 30 2014 *)

(a(n+1) - a(n))/A027642(n) = A165161(n)/A051717(n+1).
(A164558(n) - a(n))/A027642(n) = 2's = A007395.
(a(n) - A164555(n))/A027642(n) = n - 2 = A023444(n).

A174129 Numerators of the first column of the table of fractions generated by the Akiyama-Tanigawa transform from a first row A164555(k)/A027642(k).

Original entry on oeis.org

1, 1, -1, -1, 31, 7, -1051, -201, 56911, 18311, -24346415, -4227881, 425739604981, 2082738855, -759610463437, -1935668684041, 91825384886337407, 3104887811293639, -333936446105326262497, -8039608511660213481, 496858217433153341005061

Offset: 0

Paul Curtz, Mar 09 2010

The first 6 rows if the table generated by iterative application of the Akiyama-Tanigawa transform starting with a header row of fractions  A164555(k)/A027642(k) are:
  1, 1/2, 1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0, 5/66, 0, -691/2730, 0, 7/6, ...
  1/2, 2/3, 1/2, 2/15, -1/6, -1/7, 1/6, 4/15, -3/10, -25/33, 5/6, 1382/455, ...
  -1/6, 1/3, 11/10, 6/5, -5/42, -13/7, -7/10, 68/15, 453/110, -175/11, ...
  -1/2, -23/15, -3/10, 554/105, 365/42, -243/35, -1099/30, 548/165, 19827/110, ...
  31/30, -37/15, -1171/70, -478/35, 469/6, 1247/7, -6153/22, -46708/33, ...
  7/2, 599/21, -129/14, -38566/105, -20995/42, 211515/77, 524699/66, ...
The numerators of the leftmost column define the current sequence.

D. Merlini, R. Sprugnoli and M. C. Verri, The Akiyama-Tanigawa Transformation, Integers, 5 (1) (2005) #A05.

Cf. A141056 (denominators), A174110, A174111 (first row).

Cf. A164555, A027642, A048993.

read("transforms3") ;
A174129 := proc(n) Lin := [bernoulli(0),-bernoulli(1),seq(bernoulli(k),k=2..n+1)] ; for r from 1 to n do Lin := AKIYATANI(Lin) ; end do; numer(op(1,Lin)) ; end proc:

a[0, k_] := a[0, k] = BernoulliB[k]; a[0, 1] = 1/2; a[n_, k_] := a[n, k] = (k+1)*(a[n-1, k] - a[n-1, k+1]); a[n_] := a[n, 0] // Numerator; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Aug 14 2012 *)

a(n) = numerator(Sum_{j=0..n} (-1)^(n-j)*j!*Stirling2(n,j)*B(j)), where B are the Bernoulli numbers A164555/A027642. - Fabián Pereyra, Jan 06 2022

A176184 a(2n) = A027641(n). a(2n+1) = A164555(n).

Original entry on oeis.org

1, 1, -1, 1, 1, 1, 0, 0, -1, -1, 0, 0, 1, 1, 0, 0, -1, -1, 0, 0, 5, 5, 0, 0, -691, -691, 0, 0, 7, 7, 0, 0, -3617, -3617, 0, 0

Offset: 0

Paul Curtz, Apr 11 2010

Essentially the same as A176144. (The signs of the third and fourth entry are swapped.)
This refers to a shuffling of the "original" Bernoulli numbers and the Bernoulli numbers in opposite order compared to the composition discussed in A176150.
The inverse binomial transform of the shuffle in A176150 was 1,0, -1/2, 0, 13/6, -20/3. The shuffling here would yield an inverse binomial transform 1, 0, -3/2, 4, -47/6, 40/3, -21, 95/3 etc.
The difference between the corresponding elements of these two binomial transforms element by element is 0, 0, 1, -4, 10, -20, 35, -56, 84, -120, 165, -220,..., a signed variant of A000292.

A191972 The numerators of T(n, n+1) with T(0, m) = A164555(m)/A027642(m) and T(n, m) = T(n-1, m+1) - T(n-1, m), n >= 1, m >= 0.

Original entry on oeis.org

1, -1, 1, -4, 4, -16, 3056, -1856, 181312, -35853056, 1670556928, -39832634368, 545273832448, -19385421824, 53026545299456, -2753673793480966144, 68423881271489019904, -22654998127210332160

Offset: 0

Paul Curtz, Jun 20 2011

For the denominators of T(n, n+1) see A190339, where detailed information can be found.

			T(n,n+1) = [1/2, -1/6, 1/15 , -4/105, 4/105, -16/231, 3056/15015, -1856/2145, 181312/36465, ...]

Cf. A191302, A191754, A181131.

nmax:=20: mmax:=nmax: A164555:=proc(n): if n=1 then 1 else numer(bernoulli(n)) fi: end: A027642:=proc(n): if n=1 then 2 else denom(bernoulli(n)) fi: end: for m from 0 to 2*mmax do T(0,m):=A164555(m)/A027642(m) od: for n from 1 to nmax do for m from 0 to 2*mmax do T(n,m):=T(n-1,m+1)-T(n-1,m) od: od: for n from 0 to nmax do seq(T(n,m),m=0..mmax) od: seq(numer(T(n,n+1)),n=0..nmax-1); # Johannes W. Meijer, Jun 30 2011

nmax = 17; b[n_] := BernoulliB[n]; b[1] = 1/2; bb = Table[b[n], {n, 0, 2*nmax+1}]; dd = Table[Differences[bb, n], {n, 1, nmax }]; a[0] = 1; a[n_] := dd[[n, n+2]] // Numerator; Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Oct 02 2012 *)

T(n, n+1) = T(n, n)/2.

a(n+2) = (-1)^n*A181130(n+2)/2.

Thanks to R. J. Mathar by Paul Curtz, Jun 20 2011

Edited by Johannes W. Meijer, Jun 30 2011

cp's OEIS Frontend

A181722 Numerator of (1/n - Bernoulli number A164555(n)/A027642(n)).

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A165226 Numerator of 1 - A164555(n)/A027642(n).

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A172298 a(n) = A027641(n) * A164555(n).

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A174342 Denominator of ( A164555(n)/A027642(n) + 1/(n+1) ).

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A176144 a(2n) = A164555(n). a(2n+1) = A027641(n).

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A193220 Denominators of the fourth row of Akiyama-Tanigawa algorithm leading to Bernoulli numbers A164555(n)/A027642(n).

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A235774 Let b(k) = A164555(k)/A027642(k), the sequence of "original" Bernoulli numbers with -1 instead of A164555(0)=1; then a(n) = numerator of the n-th term of the binomial transform of the b(k) sequence.

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A174129 Numerators of the first column of the table of fractions generated by the Akiyama-Tanigawa transform from a first row A164555(k)/A027642(k).

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A176184 a(2n) = A027641(n). a(2n+1) = A164555(n).

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