A027642 -id:A027642 - cp's OEIS frontend

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

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

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

A285864 Triangle read by rows: a(n,m) = numerator(binomial(n,m)2^(n-m)B(n-m)) with B(k) the Bernoulli numbers A027641(k)/A027642(k).

Original entry on oeis.org

1, -1, 1, 2, -2, 1, 0, 2, -3, 1, -8, 0, 4, -4, 1, 0, -8, 0, 20, -5, 1, 32, 0, -8, 0, 10, -6, 1, 0, 32, 0, -56, 0, 14, -7, 1, -128, 0, 128, 0, -112, 0, 56, -8, 1, 0, -384, 0, 128, 0, -336, 0, 24, -9, 1, 2560, 0, -384, 0, 320, 0, -112, 0, 30, -10, 1

Offset: 0

Wolfdieter Lang, May 03 2017

The denominator triangle b(n,m) is given in A285865.
a(n,m)/b(n,m) = B(2;n,m) is the d = 2 instance of the fractional d-family of triangles B(d;n,m) = binomial(n,m)*d^(n-m)*B(n-m), for d >= 1. They are the coefficient triangles of generalized Bernoulli polynomials PB(d;n,x) = Sum_{m=0..n} B(d;n,m)*x^m for n >= 0.
{PB(d;n,x)}{n>=0} has e.g.f. EB(d;x,z) := Sum{n>=0} PB(d;n,x)*z^n = d*z*exp(x*z)/(exp(d*z)-1). B(d;n,m) is a Sheffer triangle of the Appell type for each d, denoted by (d*z/(exp(d*z - 1)), z).
PB(d;n,x) gives a (trivial) generalization of the Bernoulli polynomials with coefficients given in A196838/A196839 (rising powers of x), and this is PB(1;n,x).
The polynomials PB(d;n,x) appear in the generalized Faulhaber formula for sums of powers of arithmetic progressions SP(n,m) := Sum_{j=0..m} (a + d*j)^n, n >= 0, m >= 0, d >= 1, a = 0 for d = 1 and a from the smallest positive restricted residue system modulo d >= 2. For this Faulhaber formula see a comment in A285863, where they are named B(d;n,x).
The row sums of the rational triangle B(2;n,m) give A157779(n)/A141459(n). The alternating row sums are given in A285866/A141459(n).

			The triangle a(n,m) begins:
n\m    0    1    2   3    4    5    6  7  8   9 10 ...
0:     1
1:    -1    1
2:     2   -2    1
3:     0    2   -3   1
4:    -8    0    4  -4    1
5:     0   -8    0  20   -5    1
6:    32    0   -8   0   10   -6    1
7:     0   32    0 -56    0   14   -7  1
8:  -128    0  128   0 -112    0   56 -8  1
9:     0 -384    0 128    0 -336    0 24 -9   1
10: 2560    0 -384   0  320    0 -112  0 30 -10  1
...
The rational triangle B(2;n,m) = a(n,m)/A285865(n,m) begins:
n\m     0       1        2     3     4      5     6    7    8   9  10 ...
0:      1
1:     -1       1
2:     2/3     -2        1
3:      0       2       -3     1
4:    -8/15     0        4    -4     1
5:      0     -8/3       0   20/3   -5      1
6:    32/21     0       -8     0    10     -6     1
7:      0     32/3       0  -56/3    0     14    -7    1
8:  -128/15     0      128/3   0  -112/3    0   56/3  -8    1
9:      0    -384/5      0    128    0   -336/5   0   24   -9   1
10:  2560/33    0      -384    0    320     0   -112   0   30 -10   1
...

Cf. A027641/A027642, A157779/A141459, A196838/A196839, A285863, A285865.

T := d -> (n,m) -> numer(binomial(n, m)*d^(n-m)*bernoulli(n-m)):
for n from 0 to 10 do seq(T(2)(n,k),k=0..n) od; # Peter Luschny, May 04 2017

T[n_, m_]:=Numerator[Binomial[n, m]*2^(n - m)*BernoulliB[n - m]]; Table[T[n, m], {n, 0, 20}, {m, 0, n}] // Flatten (* Indranil Ghosh, May 06 2017 *)

T(n, m) = numerator(binomial(n, m)*2^(n - m)*bernfrac(n - m));
for(n=0, 20, for(m=0, n, print1(T(n, m),", ");); print();) \\ Indranil Ghosh, May 06 2017

from sympy import binomial, bernoulli
def T(n, m): return (binomial(n, m) * (-2)**(n - m) * bernoulli(n - m)).numerator
for n in range(21): print([T(n, m) for m in range(n + 1)]) # Indranil Ghosh, May 06 2017

a(n,m) = numerator(binomial(n, m)*2^(n-m)*B(n-m)), with the Bernoulli numbers B(k) = A027641(k)/A027642(k).

E.g.f.s of the rational column sequences {B(2;n, m)}_{n>=0} are Ecol(m, x) = (2*x/(exp(2*x) - 1))*x^m/m! (Sheffer property). Here the numerators of column m are numerator([x^m/m!] Ecol(m, x)), m >= 0.

A166333 The largest prime that divides A027642(n) (the denominator of the Bernoulli number B_n), or 1 if A027642(n) is 1.

Original entry on oeis.org

1, 2, 3, 1, 5, 1, 7, 1, 5, 1, 11, 1, 13, 1, 3, 1, 17, 1, 19, 1, 11, 1, 23, 1, 13, 1, 3, 1, 29, 1, 31, 1, 17, 1, 3, 1, 37, 1, 3, 1, 41, 1, 43, 1, 23, 1, 47, 1, 17, 1, 11, 1, 53, 1, 19, 1, 29, 1, 59, 1, 61, 1, 3, 1, 17, 1, 67, 1, 5, 1, 71, 1, 73, 1, 3, 1, 5, 1, 79, 1, 41, 1, 83, 1, 43, 1, 3, 1, 89, 1

Offset: 0

Paul Curtz, Oct 12 2009

nonn

The largest member of the extended prime list A008578 which divides the denominator of Bernoulli(n).

Essentially A073409 padded with 1's.

Antti Karttunen, Table of n, a(n) for n = 0..16384

Cf. A006530, A027642, A073409, A089026, A166120.

A006530(n) = if(n>1, vecmax(factor(n)[, 1]), 1);
A166333(n) = A006530(denominator(bernfrac(n))); \\ Antti Karttunen, Dec 19 2018

a(n) = A006530(A027642(n)). - Antti Karttunen, Dec 19 2018

Edited and extended by R. J. Mathar, Oct 21 2009

Name and comment swapped by Antti Karttunen, Dec 19 2018

A172032 Numerator of the rational sequence c(n) defined by c(n+1) - 2*c(n) = Bernoulli number B_n (A027641/A027642).

Original entry on oeis.org

0, 1, 3, 19, 19, 379, 379, 3539, 3539, 42461, 42461, 1868459, 1868459, 32384089, 32384089, 388644103, 388644103, 26424178387, 26424178387, 669590253599, 669590253599, 1605990140413, 1605990140413, 148027376624695, 148027376624695, 980410698447157

Offset: 0

Paul Curtz, Jan 23 2010

c(n) starts with: 0, 1, 3/2, 19/6, 19/3,3 79/30, 379/15, 3539/70, 3539/35, 42461/210, 42461/105, ...
The corresponding denominator is A172031 (also denominator of rational sequence defined in A172030).
It appears that A172030/A172031 - A172032/A172031 = 0, 0, 1, 2, 4, 8, 16, ... that is A131577 prepended with 0.

aseq(m) = {cvec = vector(m); cvec[1] = 0; for (i=2, m, cvec[i] = bernfrac(i-2) + 2*cvec[i-1];);} \\Michel Marcus, Feb 03 2013

Edited by Michel Marcus, Feb 03 2013

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

A216639 A027642(6*n+6)/(sequence of period 2:repeat 42,210).

Original entry on oeis.org

1, 13, 19, 13, 341, 9139, 43, 221, 19, 270413, 1541, 667147, 79, 16211, 6479, 21437, 103, 996151, 1, 11086933, 103759, 20033, 6533, 11341499, 51491, 8545667, 3097, 16211, 59, 34408161359, 1, 4137341, 5826521, 1339, 219666403, 72719023, 223, 2977, 1501, 45423164501, 83

Offset: 0

Paul Curtz, Sep 12 2012

nonn

Is a(n) always an integer?  Is there an a(n) ending with 5?
It appears (tested for n <= 800) that a(n) mod 9 is always one of {1, 2, 4, 5, 7, 8}.
There is a similar sequence of ratios A027642(10n+1)/(66*A010686(n)) which starts 1, 1, 217, 41, 1, 172081, 71, 697, 4123, 101, 23, 7055321, 131, 2059, 32767, 697, 1, 21896102683,...
a(n) is always an integer: 42 = 2*3*7 and 1, 2, and 6 divide 12n+6; 210 = 2*3*5*7 and 1, 2, 4, and 6 divide 12n+12. a(n) never ends in 5 (or 0) since 12n+6 is not divisible by 4 hence the (12n+6)-th Bernoulli denominator is not divisible by 5, and Bernoulli denominators are squarefree and hence the (12n+12)-th Bernoulli denominator, divided by 210, cannot be divisible by 5. - Charles R Greathouse IV, Sep 12 2012
The previous comments argue that 3 or 5 are never prime divisors of a(n). In addition (tested up to n <=900), 7 apparently is also a non-divisor of a(n). In summary, the prime divisors appear all to be in A140461. - Jean-François Alcover, Sep 17 2012

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000
A. Joyal, Les nombres de Bernoulli (in French), 2003.

Cf. A002445, A027762, A165734, A165949.

A010686 := proc(n)
    op((n mod 2)+1,[1,5]) ;
end proc:
A216639 := proc(n)
    A027642(6*n+6)/42/A010686(n) ;
end proc: # R. J. Mathar, Sep 21 2012

a(n)=denominator(bernfrac(6*n+6))/if(n%2,210,42) \\ Charles R Greathouse IV, Sep 12 2012

a(n)=my(t=1);fordiv(3*n+3,d,if(isprime(2*d+1),t*=2*d+1));t/if(n%2,105,21) \\ Charles R Greathouse IV, Sep 12 2012

a(n) = A027642(6*n+6)/(42*A010686(n)).

a(20)-a(40) from Charles R Greathouse IV, Sep 12 2012

A227500 a(0)=a(1)=0; for n>1, a(n) = numerator( r(n) ), where r(n) = r(n-1)+r(n-2)+A027641(n-2)/A027642(n-2) and r(0)=r(1)=a(0).

Original entry on oeis.org

0, 0, 1, 1, 5, 13, 19, 179, 1028, 1103, 893, 2889, 15445, 249787, 24988, 8494711, 6888613, 7423979, 101535859, 329279361, 1187585188, 128951009, 2513033741, 25007430139, 599126628077, 591141383117, -3361274604, 1470023540617, 22712552603063, 322385807064733, -26340115994784101

Offset: 0

Paul Curtz, Jul 13 2013

Reduced a(n)/c(n) = 0, 0, 1, 1/2, 5/3, 13/6, 19/5, 179/30, 1028/105, 1103/70, 893/35, 2889/70, 15445/231, 249787/2310,... .
After the first Bernoulli numbers we consider the same transform applied to the second Bernoulli numbers A164555(n)/A027642(n). Hence reduced b(n)/c(n) = 0, 0, 1, 3/2, 8/3, 25/6, 34/5, 329/30, 1868/105, 2013/70, 1628/35, 5269/70, 28150/231, 455377/2310, ....
Conjecture: (b(n)-a(n))/c(n) = 0, 0, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ..., that is two 0 followed by A000045.
This conjecture is confirmed up to 100 terms. [Jean-François Alcover, Jul 19 2013]

			a(2)=1 because r(2)=r(1)+r(0)+A027641(0)/A027642(0)=0+0+1=1;
a(3)=1 because r(3)=r(2)+r(1)+A027641(1)/A027642(1)=1+0-1/2=1/2;
a(4)=5 because r(4)=r(3)+r(2)+A027641(2)/A027642(2)=1+1/2+1/6=5/3.

b1[0] = b1[1] = 0; b1[n_] := b1[n] = b1[n - 1] + b1[n - 2] + BernoulliB[n - 2]; a[n_] := Numerator[b1[n]]; Table[a[n], {n, 0, 30}]  (* Jean-François Alcover, Jul 19 2013 *)

More terms from Jean-François Alcover, Jul 19 2013

cp's OEIS Frontend

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

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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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A285864 Triangle read by rows: a(n,m) = numerator(binomial(n,m)*2^(n-m)*B(n-m)) with B(k) the Bernoulli numbers A027641(k)/A027642(k).

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A166333 The largest prime that divides A027642(n) (the denominator of the Bernoulli number B_n), or 1 if A027642(n) is 1.

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A172032 Numerator of the rational sequence c(n) defined by c(n+1) - 2*c(n) = Bernoulli number B_n (A027641/A027642).

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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.

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A285864 Triangle read by rows: a(n,m) = numerator(binomial(n,m)2^(n-m)B(n-m)) with B(k) the Bernoulli numbers A027641(k)/A027642(k).