A027651 -id:A027651 - cp's OEIS frontend

A099594 Array read by antidiagonals: poly-Bernoulli numbers B(-k,n).

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 8, 14, 8, 1, 1, 16, 46, 46, 16, 1, 1, 32, 146, 230, 146, 32, 1, 1, 64, 454, 1066, 1066, 454, 64, 1, 1, 128, 1394, 4718, 6902, 4718, 1394, 128, 1, 1, 256, 4246, 20266, 41506, 41506, 20266, 4246, 256, 1, 1, 512, 12866, 85310, 237686, 329462, 237686, 85310, 12866, 512, 1

Offset: 0

Ralf Stephan, Oct 27 2004

B_n^{(-k)} is the number of distinct n by k "lonesum matrices" where a matrix of entries 0 or 1 is called lonesum when it is uniquely reconstructible from its row and column sums. [Brewbaker]
B_n^{(-k)} is the cardinality of the set { sigma in S_{n+k}: -k <= i-sigma(i) <= n for all i=1,2,...,n+k }. [Launois]
T(n,k) is also the number of permutations on [n+k] in which each substring whose support belongs to {1, 2, ..., n} or {n+1, n+2, ..., n+k} is increasing. For example, with n = 2 and k = 3, the permutation 41532 does not qualify because the substring 53 has support in {n+1, n+2, ..., n+k} = {3,4,5} but is not increasing. T(2,1) = 4 counts 123, 132, 231, 312 while the permutations satisfying Launois' condition above are 123, 132, 213, 231. A bijection between these sets of permutations would be interesting. - David Callan, Jul 22 2008. (Corrected by Norman Do, Sep 01 2008)
T(n,k) is also the number of acyclic orientations of the complete bipartite graph K_{n,k}. - Vincent Pilaud, Sep 15 2020
When indexed as a triangular array, T(n,k) is the number of permutations of [n] in which 1 is in position k and the excedance entries are precisely the entries to the left of 1. See link. - David Callan, Dec 12 2021
T(n,k) is also the number of max-closed relations between an ordered n-element set and an ordered k-element set (see the paper by Jeavons and Cooper 1995). -  Don Knuth, Feb 12 2024

			Square array begins:
  1,  1,   1,    1,     1,      1, ...
  1,  2,   4,    8,    16,     32, ...
  1,  4,  14,   46,   146,    454, ...
  1,  8,  46,  230,  1066,   4718, ...
  1, 16, 146, 1066,  6902,  41506, ...
  1, 32, 454, 4718, 41506, 329462, ...
  ...

Alois P. Heinz, Rows n = 0..140, flattened
Arvind Ayyer and Beáta Bényi, Toppling on permutations with an extra chip, arXiv:2104.13654 [math.CO], 2021. See Table 1 (a) p. 4.
Beáta Bényi, Advances in Bijective Combinatorics, Ph. D. Dissertation, Doctoral School of Mathematics and Computer Science, University of Szeged, Bolyai Institute, 2014. See Table 1.
Beáta Bényi and Peter Hajnal, Combinatorics of poly-Bernoulli numbers, arXiv:1510.05765 [math.CO], 2015.
Beata Bényi and Peter Hajnal, Combinatorial properties of poly-Bernoulli relatives, arXiv preprint arXiv:1602.08684 [math.CO], 2016.
Beata Benyi and Peter Hajnal, Poly-Bernoulli Numbers and Eulerian Numbers, arXiv:1804.01868 [math.CO], 2018.
Beáta Bényi and Matthieu Josuat-Vergès, Combinatorial proof of an identity on Genocchi numbers, arXiv:2010.10060 [math.CO], 2020.
Beáta Bényi and Gábor V. Nagy, Bijective enumerations of Γ-free 0-1 matrices, arXiv:1707.06899 [math.CO], 2017.
Beáta Bényi and José Luis Ramírez, On q-poly-Bernoulli numbers arising from combinatorial interpretations, arXiv:1909.09949 [math.CO], 2019.
Beáta Bényi and José Luis Ramírez, Poly-Cauchy numbers - the combinatorics behind, arXiv:2105.04791 [math.CO], 2021.
Beáta Bényi and José Luis Ramírez, Poly-Cauchy Numbers of the Second Kind-the Combinatorics Behind, Enumerative Comb. Appl. (2022) Vol. 2, No. 1, Art. #S2R1.
Chad Brewbaker, A combinatorial interpretation of the poly-Bernoulli numbers and two Fermat analogues, INTEGERS Vol. 8 (2008), #A02.
David Callan, Permutations whose excedance positions are those before 1
Frédéric Chapoton and Vincent Pilaud, Shuffles of deformed permutahedra, multiplihedra, constrainahedra, and biassociahedra, arXiv:2201.06896 [math.CO], 2022. See p. 12.
Peter G. Jeavons and Martin C. Cooper, Tractable constraints on ordered domains, Artificial Intelligence 79 (1995), 327-339.
Hyeong-Kwan Ju and Seunghyun Seo, Enumeration of (0,1)-matrices avoiding some 2 X 2 matrices, Discrete Math., 312 (2012), 2473-2481.
Ken Kamano, Lonesum decomposable matrices, arXiv:1701.07157 [math.CO], 2017.
Masanobu Kaneko, Poly-Bernoulli numbers, Journal de théorie des nombres de Bordeaux, 9 no. 1 (1997), Pages 221-228.
Anatol N. Kirillov, On some quadratic algebras. I 1/2: Combinatorics of Dunkl and Gaudin elements, Schubert, Grothendieck, Fuss-Catalan, universal Tutte and reduced polynomials, SIGMA, Symmetry Integrability Geom. Methods Appl. 12, Paper 002, 172 p. (2016).
Don Knuth, Parades and poly-Bernoulli bijections, Mar 31 2024. See (0.1).
D. E. Knuth, Notes on four arrays of numbers arising from the enumeration of CRC constraints and min-and-max-closed constraints, May 06 2024. Mentions this sequence.
Stéphane Launois, Combinatorics of H-primes in quantum matrices, Journal of Algebra, Volume 309, Issue 1, 2007, Pages 139-167.
Eric Weisstein's World of Mathematics, Complete Bipartite Graph
H. A. Witek, G. Mos and C.-P. Chou, Zhang-Zhang Polynomials of Regular 3-and 4-tier Benzenoid Strips, MATCH Commun. Math. Comput. Chem. 73 (2015) 427-442.
Wikipedia, Acyclic orientation

Rows 0..9 are A000012, A000079, A027649, A027650, A027651, A283811, A283812, A283813, A284032, A284033.
Main diagonal is A048163. Another diagonal is A188634.
Antidiagonal sums are in A098830.
Cf. A019538, A210381, A266695.

A:= (n, k)-> add(Stirling2(n+1, i+1)*Stirling2(k+1, i+1)*
             i!^2, i=0..min(n, k)):
seq(seq(A(n, d-n), n=0..d), d=0..10);  # Alois P. Heinz, Jan 02 2016

T[n_, k_] := Sum[(-1)^(j+n)*(1+j)^k*j!*StirlingS2[n, j], {j, 0, n}]; Table[ T[n-k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 30 2016 *)

T(n,k)=sum(j=0,n,(j+1)^k*sum(i=0,j,(-1)^(n+j-i)*binomial(j,i)*(j-i)^n))

T(n,k)=sum(j=0,min(n,k), j!^2*stirling(n+1, j+1, 2)*stirling(k+1, j+1, 2)); \\ Michel Marcus, Mar 05 2017

pB(k, n) = (-1)^n * Sum[i=0..n, (-1)^i * i! * Stirling2(n, i) / (i+1)^k ].
E.g.f.: e^(x+y) / [e^x + e^y - e^(x+y)].
T(n, k) = Sum_{j=0..n} (j+1)^k*Sum_{i=0..j} (-1)^(n+j-i)*C(j, i)*(j-i)^n. - Paul D. Hanna, Nov 04 2004
n-th row of the array = row sums of n-th power of triangle A210381. - Gary W. Adamson, Mar 21 2012

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

Original entry on oeis.org

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

A027648 Denominators of poly-Bernoulli numbers B_n^(k) with k=4.

Original entry on oeis.org

1, 16, 1296, 3456, 3240000, 144000, 1555848000, 59270400, 5000940000, 1587600, 9762501672000, 11269843200, 221794053611130000, 39390663312000, 5849513501832000, 519437318400, 407131014322092060000, 1063331477208000

Offset: 0

N. J. A. Sloane

Seiichi Manyama, Table of n, a(n) for n = 0..807
K. Imatomi, M. Kaneko, and 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. A027641, A027642, A027643, A027644, A027645, A027646, A027647, A027649, A027650, A027651.

A027648:= func< n,k | Denominator( (&+[(-1)^(j+n)*Factorial(j)*StirlingSecond(n, j)/(j+1)^k: j in [0..n]]) ) >;
[A027648(n,4): n in [0..20]]; // G. C. Greubel, Aug 02 2022

a:= (n, k) -> denom( (-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[Denominator@ Sum[((-1)^(m + n))*m!*StirlingS2[n, m]*(m + 1)^(-k), {m, 0, n}], {n, 0, 17}]] (* Michael De Vlieger, Mar 18 2017 *)

def A027648(n,k): return denominator( sum((-1)^(n+j)*factorial(j)*stirling_number2(n,j)/(j+1)^k for j in (0..n)) )
[A027648(n,4) for n in (0..20)] # G. C. Greubel, Aug 02 2022

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

A372261 Number T(n,k,j) of acyclic orientations of the complete tripartite graph K_{n,k,j}; triangle of triangles T(n,k,j), n>=0, k=0..n, j=0..k, read by rows.

Original entry on oeis.org

1, 1, 2, 6, 1, 4, 18, 14, 78, 426, 1, 8, 54, 46, 330, 2286, 230, 1902, 15402, 122190, 1, 16, 162, 146, 1374, 12090, 1066, 10554, 101502, 951546, 6902, 76110, 822954, 8724078, 90768378, 1, 32, 486, 454, 5658, 63198, 4718, 57054, 657210, 7290942, 41506, 525642, 6495534, 78463434, 928340190

Offset: 0

Alois P. Heinz, Apr 24 2024

An acyclic orientation is an assignment of a direction to each edge such that no cycle in the graph is consistently oriented. Stanley showed that the number of acyclic orientations of a graph G is equal to the absolute value of the chromatic polynomial X_G(q) evaluated at q=-1.

			Triangle of triangles T(n,k,j) begins:
    1;
  ;
    1;
    2,    6;
  ;
    1;
    4,   18;
   14,   78,   426;
  ;
    1;
    8,   54;
   46,  330,  2286;
  230, 1902, 15402, 122190;
  ;
  ...

Alois P. Heinz, Rows n = 0..40, flattened
Don Knuth, Parades and poly-Bernoulli bijections, Mar 31 2024. See (19.2).
Richard P. Stanley, Acyclic Orientations of Graphs, Discrete Mathematics, 5 (1973), pages 171-178, doi:10.1016/0012-365X(73)90108-8
Wikipedia, Acyclic orientation
Wikipedia, Multipartite graph

T(n,k,0) for k=0..9 give: A000012, A000079, A027649, A027650, A027651, A283811, A283812, A283813, A284032, A284033.
T(n,n,n) gives A370961.
T(n,n,0) gives A048163(n+1).
T(n+1,n,0) gives A188634(n+1).
T(n,1,1) gives A008776.
T(n,2,2) gives A370960.
Cf. A099594, A212220, A267383, A372254.

g:= proc(n) option remember; `if`(n=0, 1, add(
      expand(x*g(n-j))*binomial(n-1, j-1), j=1..n))
    end:
T:= proc() option remember; local q, l, b; q, l, b:= -1, [args],
      proc(n, j) option remember; `if`(j=1, mul(q-i, i=0..n-1)*
        (q-n)^l[1], add(b(n+m, j-1)*coeff(g(l[j]), x, m), m=0..l[j]))
      end; abs(b(0, nops(l)))
    end:
seq(seq(seq(T(n, k, j), j=0..k), k=0..n), n=0..5);

g[n_] := g[n] = If[n == 0, 1, Sum[Expand[x*g[n - j]]*Binomial[n - 1, j - 1], {j, 1, n}]];
T[n_, k_, j_] := T[n, k, j] = Module[{q = -1, l = {n, k, j}, b},
   b[n0_, j0_] := b[n0, j0] = If[j0 == 1, Product[q - i, {i, 0, n0 - 1}]*
   (q - n0)^n, Sum[b[n0 + m, j0 - 1]*Coefficient[g[l[[j0]]], x, m],
   {m, 0, l[[j0]]}]];
Abs[b[0, 3]]];
Table[Table[Table[T[n, k, j], {j, 0, k}], {k, 0, n}], {n, 0, 5}] // Flatten (* Jean-François Alcover, Jun 14 2024, after Alois P. Heinz *)

A329718 The number of open tours by a biased rook on a specific f(n) X 1 board, where f(n) = A070941(n) and cells are colored white or black according to the binary representation of 2n.

Original entry on oeis.org

1, 2, 4, 4, 8, 6, 14, 8, 16, 10, 24, 10, 46, 24, 46, 16, 32, 18, 44, 14, 84, 34, 68, 18, 146, 68, 138, 44, 230, 84, 146, 32, 64, 34, 84, 22, 160, 54, 112, 22, 276, 106, 224, 54, 376, 106, 192, 34, 454, 192, 406, 112, 690, 224, 406, 84, 1066, 376, 690, 160

Offset: 0

Mikhail Kurkov, Nov 19 2019 [verification needed]

nonn

A cell is colored white if the binary digit is 0 and a cell is colored black if the binary digit is 1. A biased rook on a white cell moves only to the left and otherwise moves only to the right.

			a(1) = 2 because the binary expansion of 2 is 10 and there are 2 open biased rook's tours, namely 12 and 21.
a(2) = 4 because the binary expansion of 4 is 100 and there are 4 open biased rook's tours, namely 132, 213, 231 and 321.
a(3) = 4 because the binary expansion of 6 is 110 and there are 4 open biased rook's tours, namely 123, 132, 231 and 312.

Mikhail Kurkov, Comments on A329718 [verification needed]

Cf. A027649, A027650, A027651, A059894, A110501, A283811, A284005, A329369, A344902, A344947.

a(n) = f(n) + f(A059894(n)) = f(n) + f(2*A053645(n)) for n > 0 with a(0) = 1 where f(n) = A329369(n).
Sum_{k=0..2^n-1} a(k) = 2*(n+1)! - 1 for n >= 0.
a((4^n-1)/3) = 2*A110501(n+1) for n > 0.
a(2^1*(2^n-1)) = A027649(n),
a(2^2*(2^n-1)) = A027650(n),
a(2^3*(2^n-1)) = A027651(n),
a(2^4*(2^n-1)) = A283811(n),
and more generally, a(2^m*(2^n-1)) = T(n,m+1) for n >= 0, m >= 0 where T(n,m) = Sum_{k=0..n} k!*(k+1)^m*Stirling2(n,k)*(-1)^(n-k).

cp's OEIS Frontend

A099594 Array read by antidiagonals: poly-Bernoulli numbers B(-k,n).

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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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A027648 Denominators of poly-Bernoulli numbers B_n^(k) with k=4.

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A372261 Number T(n,k,j) of acyclic orientations of the complete tripartite graph K_{n,k,j}; triangle of triangles T(n,k,j), n>=0, k=0..n, j=0..k, read by rows.

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A329718 The number of open tours by a biased rook on a specific f(n) X 1 board, where f(n) = A070941(n) and cells are colored white or black according to the binary representation of 2n.

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