A027648 -id:A027648 - cp's OEIS frontend

A027643 Numerators of poly-Bernoulli numbers B_n^(k) with k=2.

1, 1, -1, -1, 7, 1, -38, -5, 11, 7, -3263, -15, 13399637, 7601, -8364, -91, 1437423473, 3617, -177451280177, -745739, 166416763419, 3317609, -17730427802974, -5981591, 51257173898346323, 5436374093, -107154672791057, -213827575

Offset: 0

N. J. A. Sloane

Seiichi Manyama, Table of n, a(n) for n = 0..521
K. Imatomi, M. Kaneko, and E. Takeda, Multi-Poly-Bernoulli Numbers and Finite Multiple Zeta Values, J. Int. Seq. 17 (2014) # 14.4.5
Masanobu Kaneko, Poly-Bernoulli numbers, Journal de Théorie des Nombres de Bordeaux, 9 no. 1 (1997), Pages 221-228.
Masanobu Kaneko, Poly-Bernoulli numbers, Journal de Théorie des Nombres de Bordeaux, 9 no. 1 (1997), Pages 221-228.
Index entries for sequences related to Bernoulli numbers.

Cf. A027641, A027642, A027644, A027645, A027646, A027647, A027648, A027649, A027650, A027651.

A027643:= func< n,k | Numerator( (&+[(-1)^(j+n)*Factorial(j)*StirlingSecond(n, j)/(j+1)^k: j in [0..n]]) ) >;
[A027643(n,2): n in [0..30]]; // G. C. Greubel, Aug 02 2022

a := n -> numer((-1)^n*add( (-1)^m*m!*Stirling2(n, m)/(m+1)^2, m=0..n)):
seq(a(n), n=0..27);

k=2; Table[Numerator[(-1)^n Sum[(-1)^m m! StirlingS2[n, m]/(m+1)^k, {m, 0, n}]], {n, 0, 27}] (* Michael De Vlieger, Oct 28 2015 *)

def A027643(n,k): return numerator( sum((-1)^(n+j)*factorial(j)*stirling_number2(n,j)/(j+1)^k for j in (0..n)) )
[A027643(n,2) for n in (0..30)] # G. C. Greubel, Aug 02 2022

a(n) = numerator of Sum_{k=0..n} W(n,k)*h(k+1) with W(n,k) = (-1)^(n-k)*k!* Stirling2(n+1,k+1) the Worpitzky numbers and h(n) = Sum_{k=1..n} 1/k^2 the generalized harmonic numbers of order 2. - Peter Luschny, Sep 28 2017

A027650 Poly-Bernoulli numbers B_n^(k) with k=-3.

Original entry on oeis.org

1, 8, 46, 230, 1066, 4718, 20266, 85310, 354106, 1455278, 5938186, 24104990, 97478746, 393095438, 1581931306, 6356390270, 25511588986, 102304505198, 409992599626, 1642294397150, 6576150108826, 26325519044558, 105364834103146, 421647614381630, 1687155299822266

Offset: 0

N. J. A. Sloane

a(n) is also the number of acyclic orientations of the complete bipartite graph K_{3,n}. - Vincent Pilaud, Sep 15 2020

Vincenzo Librandi, Table of n, a(n) for n = 0..500
K. Imatomi, M. Kaneko, and E. Takeda, Multi-Poly-Bernoulli Numbers and Finite Multiple Zeta Values, J. Int. Seq. 17 (2014) # 14.4.5.
Ken Kamano, Sums of Products of Bernoulli Numbers, Including Poly-Bernoulli Numbers, J. Int. Seq. 13 (2010), 10.5.2.
Ken Kamano, Sums of Products of Poly-Bernoulli Numbers of Negative Index, Journal of Integer Sequences, Vol. 15 (2012), #12.1.3.
Masanobu Kaneko, Poly-Bernoulli numbers, Journal de théorie des nombres de Bordeaux, 9 no. 1 (1997), Pages 221-228.
Takao Komatsu, Some recurrence relations of poly-Cauchy numbers, J. Nonlinear Sci. Appl., (2019) Vol. 12, Issue 12, 829-845.
Index entries for sequences related to Bernoulli numbers.
Index entries for linear recurrences with constant coefficients, signature (9,-26,24).

Cf. A027641, A027642, A027643, A027644, A027645, A027646, A027647, A027648, A027649, A027651.
First differences of A016269.
Row 3 of array A099594.

[6*4^n-6*3^n+2^n: n in [0..30]]; // Vincenzo Librandi, Jul 17 2011

a:= (n, k) -> (-1)^n*sum((-1)^j*j!*Stirling2(n, j)/(j+1)^k, j=0..n);
seq(a(n, -3), n = 0..30);

Table[6*4^n-6*3^n+2^n, {n,0,30}] (* G. C. Greubel, Feb 07 2018 *)

Vec((1-x)/((1-2*x)*(1-3*x)*(1-4*x)) + O(x^30)) \\ Michel Marcus, Feb 13 2015

[2^n -6*3^n +6*4^n for n in (0..30)] # G. C. Greubel, Aug 02 2022

a(n) = 6*4^n - 6*3^n + 2^n. - Vladeta Jovovic, Nov 14 2003
G.f.: (1-x)/((1-2*x)*(1-3*x)*(1-4*x)).
E.g.f.: exp(2*x) - 6*exp(3*x) + 6*exp(4*x). - G. C. Greubel, Aug 02 2022

A027651 Poly-Bernoulli numbers B_n^(k) with k=-4.

Original entry on oeis.org

1, 16, 146, 1066, 6902, 41506, 237686, 1315666, 7107302, 37712866, 197451926, 1023358066, 5262831302, 26903268226, 136887643766, 693968021266, 3508093140902, 17693879415586, 89084256837206, 447884338361266, 2249284754708102, 11285908565322946, 56587579617416246

Offset: 0

N. J. A. Sloane

a(n) is also the number of acyclic orientations of the complete bipartite graph K_{4,n}. - Vincent Pilaud, Sep 16 2020

Vincenzo Librandi, Table of n, a(n) for n = 0..500
K. Imatomi, M. Kaneko, and E. Takeda, Multi-Poly-Bernoulli Numbers and Finite Multiple Zeta Values, J. Int. Seq. 17 (2014) # 14.4.5.
Ken Kamano, Sums of Products of Bernoulli Numbers, Including Poly-Bernoulli Numbers, J. Int. Seq. 13 (2010), 10.5.2.
Ken Kamano, Sums of Products of Poly-Bernoulli Numbers of Negative Index, Journal of Integer Sequences, Vol. 15 (2012), #12.1.3.
Masanobu Kaneko, Poly-Bernoulli numbers, Journal de théorie des nombres de Bordeaux, 9 no. 1 (1997), Pages 221-228.
Hiroyuki Komaki, Relations between Multi-Poly-Bernoulli numbers and Poly-Bernoulli numbers of negative index, arXiv:1503.04933 [math.NT], 2015.
Index entries for sequences related to Bernoulli numbers.
Index entries for linear recurrences with constant coefficients, signature (14,-71,154,-120).

Cf. A027641, A027642, A027643, A027644, A027645, A027646, A027647, A027648, A027649, A027650.

Row n=4 of array A099594.

[24*5^n-36*4^n+14*3^n-2^n: n in [0..30]]; // Vincenzo Librandi, Jul 17 2011

a:= (n, k) -> (-1)^n*sum((-1)^j*j!*Stirling2(n,j)/(j+1)^k, j=0..n);
seq(a(n, -4), n=0..30);

Table[24*5^n -36*4^n +14*3^n -2^n, {n,0,30}] (* G. C. Greubel, Feb 07 2018 *)
LinearRecurrence[{14,-71,154,-120},{1,16,146,1066},30] (* Harvey P. Dale, Nov 20 2019 *)

Vec((1+4*x)*((1-x)^2)/((1-2*x)*(1-3*x)*(1-4*x)*(1-5*x)) + O(x^30)) \\ Michel Marcus, Feb 13 2015

[24*5^n -36*4^n +14*3^n -2^n for n in (0..30)] # G. C. Greubel, Aug 02 2022

a(n) = 24*5^n -36*4^n +14*3^n -2^n. - Vladeta Jovovic, Nov 14 2003
G.f.: (1+4*x)*(1-x)^2/((1-2*x)*(1-3*x)*(1-4*x)*(1-5*x)).
E.g.f.: 24*exp(5*x) - 36*exp(4*x) + 14*exp(3*x) - exp(2*x). - G. C. Greubel, Feb 07 2018

A027647 Numerators of poly-Bernoulli numbers B_n^(k) with k=4.

Original entry on oeis.org

1, 1, -49, 41, 26291, -1921, 845233, 1048349, -60517579, -50233, 506605371959, 823605863, -53797712101337483, -7784082036337, 8049010408144441, 246319059461, -3910018782537447618421, 1090400590625849

Offset: 0

N. J. A. Sloane

Seiichi Manyama, Table of n, a(n) for n = 0..387
K. Imatomi, M. Kaneko, E. Takeda, Multi-Poly-Bernoulli Numbers and Finite Multiple Zeta Values, J. Int. Seq. 17 (2014) # 14.4.5
M. Kaneko, Poly-Bernoulli numbers
Masanobu Kaneko, Poly-Bernoulli numbers, Journal de théorie des nombres de Bordeaux, 9 no. 1 (1997), Pages 221-228.
Index entries for sequences related to Bernoulli numbers.

Cf. A027648.

A027647:= func< n,k | Numerator( (&+[(-1)^(j+n)*Factorial(j)*StirlingSecond(n, j)/(j+1)^k: j in [0..n]]) ) >;
[A027647(n,4): n in [0..20]]; // G. C. Greubel, Aug 02 2022

a:= (n, k)-> numer((-1)^n*add((-1)^m*m!*Stirling2(n, m)/(m+1)^k, m=0..n)):
seq(a(n, 4), n = 0..30);

With[{k = 4}, Table[Numerator@ Sum[((-1)^(m + n))*m!*StirlingS2[n, m]*(m + 1)^(-k), {m, 0, n}], {n, 0, 17}]] (* Michael De Vlieger, Mar 18 2017 *)

def A027647(n,k): return numerator( sum((-1)^(n+j)*factorial(j)*stirling_number2(n,j)/(j+1)^k for j in (0..n)) )
[A027647(n,4) for n in (0..20)] # G. C. Greubel, Aug 02 2022

a(n) = numerator of Sum_{j=0..n} (-1)^(n+j) * j! * Stirling2(n, j) * (j+1)^(-k), for k = 4.

A283926 Denominators of poly-Bernoulli numbers B_n^(k) with k=7.

Original entry on oeis.org

1, 128, 279936, 5971968, 699840000000, 93312000000, 115269666624000000, 35129803161600000, 160060165655040000000, 1016255020032000000, 103970660613603049728000000, 240047701272387993600000, 41516393959179372527058885120000000

Offset: 0

Seiichi Manyama, Mar 18 2017

			B_0^(7) = 1, B_1^(7) = 1/128, B_2^(7) = -1931/279936, B_3^(7) = 32459/5971968, ...

Seiichi Manyama, Table of n, a(n) for n = 0..407
Wikipedia, Poly-Bernoulli number

Cf. A027643-A027648, A283925.

B[n_]:= Sum[((-1)^(m + n))*m!*StirlingS2[n, m] * (m + 1)^(-7), {m, 0, n}]; Table[Denominator[B[n]], {n, 0, 15}] (* Indranil Ghosh, Mar 18 2017 *)

B(n) = sum(m=0, n, ((-1)^(m + n)) * m! * stirling(n, m, 2) * (m + 1)^(-7));
for(n=0, 15, print1(numerator(B(n)),", ")) \\ Indranil Ghosh, Mar 18 2017

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A027643 Numerators of poly-Bernoulli numbers B_n^(k) with k=2.

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A027650 Poly-Bernoulli numbers B_n^(k) with k=-3.

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A027651 Poly-Bernoulli numbers B_n^(k) with k=-4.

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A027647 Numerators of poly-Bernoulli numbers B_n^(k) with k=4.

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A283926 Denominators of poly-Bernoulli numbers B_n^(k) with k=7.

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