A002427 -id:A002427 - cp's OEIS frontend

A229979 Numerators of interleaved A063524(n) and A002427(n)/A006955(n).

0, 1, 1, 1, 0, -1, 0, 1, 0, -3, 0, 5, 0, -691, 0, 35, 0, -3617, 0, 43867, 0, -1222277, 0, 854513, 0, -1181820455, 0, 76977927, 0, -23749461029, 0, 8615841276005, 0, -84802531453387, 0, 90219075042845, 0

Offset: 0

Paul Curtz, Oct 05 2013

Numerators of Br(n) = 0, 1, 1, 1/2, 0, -1/6, 0, 1/6, 0, -3/10, 0, 5/6, 0, -691/210,... complementary Bernoulli numbers.
A164555(n)/A027642(n) is an autosequence of second kind. Its inverse binomial transform is the signed sequence and its main diagonal is the double of the first upper diagonal.
Br(n) is an autosequence of first kind. Its inverse binomial transform is the signed sequence and its main diagonal is A000004=0's.
Br(n) difference table:
0,       1,    1,  1/2,    0, -1/6,...
1,       0, -1/2, -1/2, -1/6,  1/6,...    =A140351(n)/A140219(n)
-1,   -1/2,    0,  1/3,  1/3,    0,...
1/2,   1/2,  1/3,    0, -1/3, -1/3,...
0,    -1/6, -1/3, -1/3,    0, 8/15,...
-1/6, -1/6,    0,  1/3, 8/15,    0,... etc.

Cf. A050925: a similar sequence, because 2*(n+1)*B(n) and (n+1)*B(n) have the same numerator.

a[0] = 0; a[1] = a[2] = 1; a[n_] := 2*n*BernoulliB[n-1] // Numerator; Table[a[n], {n, 0, 36}] (* Jean-François Alcover, Nov 25 2013 *)

a(2n)=A063524(n). a(2n+1)=A002427(n).
a(n) = numerators of n * b(n) with b(n)=0 followed by A164555(n)/A027642(n) = 0, 1, 1/2, 1/6, 0,... in A165142(n).
a(n+1) = numerators of Br(n+1) = Br(n) + A140351(n)/A140219(n), a(0)=Br(0)=0.

Cross-ref. to A050925 by Jean-François Alcover, Dec 09 2013

A232800 Denominators written by antidiagonals of interleaved A063524(n) and A002427(n)/A006955(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 2, 1, 2, 1, 6, 6, 3, 3, 6, 6, 1, 6, 3, 1, 3, 6, 1, 6, 6, 1, 3, 3, 1, 6, 6, 1, 6, 3, 3, 1, 3, 3, 6, 1, 10, 10, 15, 5, 15, 15, 5, 15, 10, 10, 1, 10, 5, 15, 15, 1, 15, 15, 5, 10, 1

Offset: 0

Paul Curtz, Nov 30 2013

The numerators of Br(n)=0, 1, 1, 1/2, 0, -1/6, 0, 1/6, 0, -3/10, 0, 5/6, 0,... are in A229979(n).
(Corresponding complementary Euler numbers Er(n):
A198631(2n+1)/A006519(2n+2)=1/2, -1/4, 1/2, -17/8, 31/2, -691/4,... =Ef2(n). (2*n+2)*Ef2(n)=1, -1, 3, -17, 155, 2073,...=-A001469(n+1),Genocchi numbers. Er(n)=interleaved (A063524(n) and -A001469(n+1)) =0, 1, 1, -1, 0, 3, 0, -17, 0, 155,... =-A226158(n).)

			1,
1, 1,
1, 1, 1,
2, 2, 2, 2,
1, 2, 1, 2, 1,
6, 6, 3, 3, 6, 6, etc.
Triangle of denominators of Br(n),complementary Bernoulli numbers,written by antidiagonals (see A229979).

A006955 Denominator of (2n+1) B_{2n}, where B_n are the Bernoulli numbers.

Original entry on oeis.org

1, 2, 6, 6, 10, 6, 210, 2, 30, 42, 110, 6, 546, 2, 30, 462, 170, 6, 51870, 2, 330, 42, 46, 6, 6630, 22, 30, 798, 290, 6, 930930, 2, 102, 966, 10, 66, 1919190, 2, 30, 42, 76670, 6, 680862, 2, 690, 38874, 470, 6, 46410, 2, 330, 42, 106, 6, 1919190

Offset: 0

Simon Plouffe and N. J. A. Sloane

Also denominators of asymptotic expansion of polygamma function psi''(z).

			(n+1)*B_n gives the sequence 1, -1/2, 1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0, 5/66, ...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 260, (6.4.13).
A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 73.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..1000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 260, (6.4.13).
L. Euler, (E393) De summis serierum numeros Bernoullianos involventium, reprinted in: Opera Omnia. Teubner, Leipzig, 1911, Series (1), Vol. 15, p. 93.
M. Kaneko, A recurrence formula for the Bernoulli numbers, Proc. Japan Acad., 71 A (1995), 192-193.
Index entries for sequences related to Bernoulli numbers.

Numerators are in A002427.

Cf. A050925/A050932, A000367/A002445, A006956, A123125.

gf := z / (1 - exp(-z)): ser := series(gf, z, 220):
seq(denom((n+1)!*coeff(ser, z, n)), n=0..108, 2); # Peter Luschny, Aug 29 2020

Denominator[Table[(2n+1)BernoulliB[2n],{n,0,60}]] (* Harvey P. Dale, Nov 03 2011 *)

a(n) = denominator((2*n+1)*bernfrac(2*n)); \\ Michel Marcus, Aug 06 2017

Apparently a(n) = denominator(Sum_{k=0..2*n-1} (-1)^(2*n-k+1)*E1(2*n, k+1)/ binomial(2*n, k+1)), where E1(n, k) denotes the first-order Eulerian numbers A123125. - Peter Luschny, Feb 17 2021

A140351 Numerator of the coefficient [x^1] of the Bernoulli twin number polynomial C(n,x).

Original entry on oeis.org

1, 0, -1, -1, -1, 1, 1, -1, -3, 3, 5, -5, -691, 691, 35, -35, -3617, 3617, 43867, -43867, -1222277, 1222277, 854513, -854513, -1181820455, 1181820455, 76977927, -76977927, -23749461029, 23749461029, 8615841276005, -8615841276005, -84802531453387, 84802531453387

Offset: 1

Paul Curtz, May 30 2008, Jun 23 2008

The Bernoulli twin number polynomials C(n,x) are defined in A129378.

			The coefficients [x^m]C(n,x) are a table of fractions:
1 ;
-1/2, 1;
-1/3, 0, 1;
-1/6, -1/2, 1/2, 1;
-1/30,-1/2, -1/2, 1, 1;
1/30, -1/6, -1,-1/3, 3/2, 1;
1/42, 1/6, -1/2, -5/3, 0, 2, 1;
-1/42, 1/6, 1/2, -7/6, -5/2, 1/2, 5/2, 1;
-1/30, -1/6, 2/3, 7/6, -7/3, -7/2, 7/6, 3, 1;
1/30, -3/10, -2/3, 2, 7/3, -21/5, -14/3, 2, 7/2, 1;
5/66, 3/10, -3/2, -2, 5, 21/5, -7, -6, 3, 4, 1; ...
This sequence here contains the numerators of the second column.

Cf. A002427, A129378, A129826.

C := proc(n,x) if n = 0 then 1; else add(binomial(n-1,j-1)* bernoulli(j,x),j=1..n) ; expand(%) ; end if ; end proc:
A140351 := proc(n) coeff(C(n,x),x,1) ; numer(%) ; end proc: seq(A140351(n),n=1..80) ; # R. J. Mathar, Nov 22 2009

b[n_, x_] := Coefficient[ Series[ t*E^(x*t)/(E^t - 1), {t, 0, n}], t, n]*n!; c[n_, x_] := Sum[ Binomial[n-1, j-1]*b[j, x], {j, 1, n}]; t[n_, m_] := Coefficient[c[n, x], x, m]; Table[t[n, 1] // Numerator, {n, 1, 34} ] (* Jean-François Alcover, Mar 04 2013 *)
Table[Sum[Binomial[n, k]*(k+1)*BernoulliB[k], {k, 0, n}], {n, 0, 30}] // Numerator (* Vaclav Kotesovec, Oct 05 2016 *)

makelist(num(sum((binomial(n,i)*(i+1)*bern(i)),i,0,n)),n,0,20); /* Vladimir Kruchinin, Oct 05 2016 */

a(n) = numerator(sum(i=0, n, binomial(n,i)*(i+1)*bernfrac(i))); \\ Michel Marcus, Oct 05 2016

a(n) = numerator(Sum_{i=0..n} binomial(n,i)*(i+1)*bernoulli(i)). - Vladimir Kruchinin, Oct 05 2016

Edited and extended by R. J. Mathar, Nov 22 2009

A006956 Denominator of (2n+1)(2n+2) B_{2n}, where B_n are the Bernoulli numbers. Also denominators of the asymptotic expansion of the polygamma function psi'''(z).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 15, 1, 5, 21, 5, 1, 21, 1, 1, 231, 5, 1, 1365, 1, 55, 21, 1, 1, 663, 11, 5, 57, 5, 1, 15015, 1, 17, 483, 1, 11, 25935, 1, 5, 21, 935, 1, 7917, 1, 23, 19437, 5, 1, 3315, 1, 55, 21, 1, 1, 191919, 253, 2465, 21, 5, 1, 1734915, 1, 1, 17157, 17, 1

Offset: 3

Simon Plouffe

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 260, (6.4.14).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 3..1000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 260, (6.4.14).

Numerators are in A076549. Cf. A006955/A002427.

Join[{1, 1}, Table[Denominator[(4 n^2 + 6 n + 2) BernoulliB[2 n]], {n, 1, 70}]] (* Vincenzo Librandi, Aug 02 2013 *)

More terms from Ralf Stephan, Oct 19 2002

A076549 Numerator of (2n+1)(2n+2) B_{2n}, where B_n are the Bernoulli numbers. Also numerators of the asymptotic expansion of the polygamma function psi'''(z).

Original entry on oeis.org

2, 3, 2, -1, 4, -3, 10, -691, 280, -10851, 438670, -1222277, 3418052, -1181820455, 1077690978, -23749461029, 137853460416080, -84802531453387, 541314450257070, -26315271553053477373, 761798417598805340

Offset: 0

Ralf Stephan, Oct 19 2002

Vincenzo Librandi, Table of n, a(n) for n = 0..100
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Abramowitz and Stegun, Handbook of mathematical functions, p. 260 (6.4.14)
Abramowitz and Stegun, Handbook of mathematical functions, p. 260 (6.4.14)

Denominators are in A006956. Cf. A002427/A006955.

Join[{2, 3}, Table[Numerator[(4 n^2 + 6 n + 2) BernoulliB[2 n]], {n, 1, 30}]] (* Vincenzo Librandi, Aug 02 2013 *)

A140219 Denominator of the coefficient [x^1] of the Bernoulli twin number polynomial C(n,x).

Original entry on oeis.org

1, 1, 2, 2, 6, 6, 6, 6, 10, 10, 6, 6, 210, 210, 2, 2, 30, 30, 42, 42, 110, 110, 6, 6, 546, 546, 2, 2, 30, 30, 462, 462, 170, 170, 6, 6, 51870, 51870, 2, 2, 330, 330, 42, 42, 46, 46, 6, 6, 6630, 6630, 22, 22, 30, 30, 798, 798, 290

Offset: 0

Paul Curtz, Jun 23 2008

See A140351 for the main part of the documentation.

Cf. A002427, A006955, A048594, A140351 (numerators).

C := proc(n, x) if n = 0 then 1; else add(binomial(n-1, j-1)* bernoulli(j, x), j=1..n) ; expand(%) ; end if ; end proc:
A140219 := proc(n) coeff(C(n, x), x, 1) ; denom(%) ; end proc:
seq(A140219(n), n=1..80) ; # R. J. Mathar, Sep 22 2011

Table[Sum[Binomial[n, k]*(k+1)*BernoulliB[k], {k, 0, n}], {n, 0, 60}] // Denominator (* Vaclav Kotesovec, Oct 05 2016 *)

makelist(denom(sum((binomial(n, i)*(i+1)*bern(i)), i, 0, n)), n, 0, 20); /* Vladimir Kruchinin, Oct 05 2016 */

a(n) = denominator(sum(i=0, n, binomial(n,i)*(i+1)*bernfrac(i))); \\ Michel Marcus, Oct 05 2016

a(n) = denominator(Sum_{i=0..n} binomial(n,i)*(i+1)*bern(i)). - Vladimir Kruchinin, Oct 05 2016

a(n) = A006955(floor(n/2)). - Georg Fischer, Nov 29 2022

A290533 Numerator of 2n(2n+1) B_{2n}, where B_n are the Bernoulli numbers.

Original entry on oeis.org

0, 1, -2, 1, -12, 25, -1382, 245, -28936, 131601, -2444554, 9399643, -4727281820, 1000713051, -332492454406, 43079206380025, -1356840503254192, 1533724275728365, -157891629318320864238, 723708496718865073, -1044330873985796488204

Offset: 0

Seiichi Manyama, Aug 05 2017

In 1997, Matiyasevich found the following identity;

(n+2) * Sum_{k=2..n-2} B_k*B_{n-k} - 2 * Sum_{k=2..n-2} binomial(n+2, k)*B_k*B_{n-k} = n*(n+1)*B_n for n > 3.

			B_n gives the sequence 1, -1/2, 1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0, 5/66, 0, -691/2730, ...
n*(n+1)*B_n gives the sequence 0, -1, 1, 0, -2/3, 0, 1, 0, -12/5, 0, 25/3, 0, -1382/35, 0, 245, 0, -28936/15, ...

Seiichi Manyama, Table of n, a(n) for n = 0..314
Y. Matiyasevich, Identities with Bernoulli numbers, 1997.
H. Pan and Z. W. Sun, New identities involving Bernoulli and Euler polynomials, arXiv:math/0407363 [math.NT], 2004.

Cf. A002427/A006955.

A290534 Denominator of 2n(2n+1) B_{2n}, where B_n are the Bernoulli numbers.

Original entry on oeis.org

1, 1, 3, 1, 5, 3, 35, 1, 15, 7, 11, 3, 91, 1, 15, 77, 85, 3, 8645, 1, 33, 1, 23, 3, 1105, 11, 15, 133, 145, 3, 31031, 1, 51, 161, 5, 33, 319865, 1, 15, 7, 7667, 3, 16211, 1, 345, 6479, 235, 3, 7735, 1, 33, 7, 53, 3, 319865, 23, 7395, 7, 295, 3, 7055321, 1, 3, 817

Offset: 0

Seiichi Manyama, Aug 05 2017

In 1997, Matiyasevich found the following identity;

(n+2) * Sum_{k=2..n-2} B_k*B_{n-k} - 2 * Sum_{k=2..n-2} binomial(n+2, k)*B_k*B_{n-k} = n*(n+1)*B_n for n > 3.

Seiichi Manyama, Table of n, a(n) for n = 0..500
Y. Matiyasevich, Identities with Bernoulli numbers, 1997.
Hao Pan and Zhi-Wei Sun, New identities involving Bernoulli and Euler polynomials, Journal of Combinatorial Theory, Series A Volume 113, Issue 1, January 2006, Pages 156-175.

Cf. A002427/A006955.

A242246 Numerators of n*A164555(n-1)/A027642(n-1).

Original entry on oeis.org

0, 1, 1, 1, 0, -1, 0, 1, 0, -3, 0, 5, 0, -691, 0, 35, 0, -3617, 0, 43867, 0, -1222277, 0, 854513, 0, -1181820455, 0, 76977927, 0, -23749461029, 0, 8615641276005, 0, -84802531453387, 0, 90219075042845, 0

Offset: 0

Paul Curtz, May 09 2014

sign

First multiplied shifted (second) Bernoulli numbers.
A164555(n-1)/A027642(n-1) = 0 followed by (A164555(n)/A027642(n)=1, 1/2, 1/6,...) = f(n) = 0, 1, 1/2, 1/6, 0,... .
f(n+1) - f(n) = A051716(n)/A051717(n).
Generally we consider a transform applied to the autosequences of first or second kind. An autosequence is a sequence which has its inverse binomial transform equal to the signed sequence. It is of the first kind if the main diagonal is A000004=0's. It is of the second kind if the main diagonal is the first upper diagonal multiplied by 2. A000045(n) is an autosequence of the first kind. A164555(n)/A027642(n) is an autosequence of the second kind. See A190339 (and A241269).
Here we apply the transform to the Bernoulli numbers A164555(n)/A027642(n).
We take n*(0 followed by A164555(n)/A027642(n)).
Hence the autosequence of first kind
TB1(n) = 0, 1, 1, 1/2, 0, -1/6, 0, 1/6, 0, -3/10, 0, 5/6, O, -691/210,.. .
a(n) are the numerators.
The first seven rows of the differencece table of TB1(n) are
0,       1,    1,  1/2,   0, - 1/6,    0,      1/6,...
1,       0, -1/2, -1/2,  -1/6,  1/6,  1/6,    -1/6,... =A140351(n+1)/b(n+1)
-1,   -1/2,    0,  1/3,   1/3,    0, -1/3,   -2/15,...
1/2,   1/2,  1/3,    0,  -1/3, -1/3,  1/5,   11/15,...
0,    -1/6, -1/3, -1/3,     0, 8/15, 8/15,    -4/5,...
-1/6, -1/6,    0,  1/3,  8/15,    0, -4/3,    -4/3,...
0,     1/6,  1/3,  1/5, -8/15, -4/3,    0, 512/105,... .
First and second upper diagonals: 1, -1/2, 1/3, -1/3, 8/15, -4/3, 512/105,... .
Sum of the antidiagonals:
0, 1, 1, 0, -1/2, 0, 1/2, 0, -5/6, 0, 13/6, 0, -49/6, 0,... .
(Note that the same transform applied to the second fractional Euler numbers A198631(n)/A006519(n+1) yields the Genocchi numbers -A226158(n)).
This transform can be continued:
TB2(n) = n*(0 followed by TB1(n)) =
0, 0, 2, 3, 2, 0, -1, 0, 4/3, 0, -3, 0, 10, 0, -691/15, 0, 280, 0,...
is an autosequence of second kind.
TB3(n) = 0, 0, 0, 6, 12, 10, 0, -7, 0, 12, 0, -33, 0, 130, 0, 691, 0,...
is apparently an integer autosequence of the first kind.

Cf. A199969 (autosequence).

a(n) = 0 followed by (A050925(n) = 1, -1, 1, 0,... ) with 1 instead of -1.

a(2n) = A063524(n). a(2n+1) = A002427(n).

cp's OEIS Frontend

A229979 Numerators of interleaved A063524(n) and A002427(n)/A006955(n).

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A232800 Denominators written by antidiagonals of interleaved A063524(n) and A002427(n)/A006955(n).

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A006955 Denominator of (2n+1) B_{2n}, where B_n are the Bernoulli numbers.

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A140351 Numerator of the coefficient [x^1] of the Bernoulli twin number polynomial C(n,x).

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A006956 Denominator of (2n+1)(2n+2) B_{2n}, where B_n are the Bernoulli numbers. Also denominators of the asymptotic expansion of the polygamma function psi'''(z).

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A076549 Numerator of (2n+1)(2n+2) B_{2n}, where B_n are the Bernoulli numbers. Also numerators of the asymptotic expansion of the polygamma function psi'''(z).

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A140219 Denominator of the coefficient [x^1] of the Bernoulli twin number polynomial C(n,x).

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A290533 Numerator of 2*n*(2*n+1) B_{2*n}, where B_n are the Bernoulli numbers.

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A290533 Numerator of 2n(2n+1) B_{2n}, where B_n are the Bernoulli numbers.

A290534 Denominator of 2n(2n+1) B_{2n}, where B_n are the Bernoulli numbers.