A027649 -id:A027649 - 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

A106516 A Pascal-like triangle based on 3^n.

Original entry on oeis.org

1, 3, 1, 9, 4, 1, 27, 13, 5, 1, 81, 40, 18, 6, 1, 243, 121, 58, 24, 7, 1, 729, 364, 179, 82, 31, 8, 1, 2187, 1093, 543, 261, 113, 39, 9, 1, 6561, 3280, 1636, 804, 374, 152, 48, 10, 1, 19683, 9841, 4916, 2440, 1178, 526, 200, 58, 11, 1, 59049, 29524, 14757, 7356, 3618, 1704, 726, 258, 69, 12, 1

Offset: 0

Paul Barry, May 05 2005

Row sums are A027649. Antidiagonal sums are A106517.
From Wolfdieter Lang, Jan 09 2015: (Start)
Alternating row sums give A025192. The A-sequence of this Riordan lower triangular matrix is [1, 1, repeat(0, )] (leading to the Pascal recurrence for T(n,k) for n >= k >= 1. The Z-sequence is [3, repeat(0, )] (leading to the recurrence T(n,0) = 3*T(n-1,0), n >= 1. For A- and Z-sequences see the W. Lang link under A006232.
The inverse of this Riordan matrix is Tinv = ((1 - 2*x)/(1 + x), x/(1 + x)) given as a signed version of A093560: Tinv(n,m) = (-1)^(n-m)*A093560(n,m). (End)

			The triangle T(n,k) begins:
n\k     0     1     2    3    4    5   6   7  8  9 10 ...
0:      1
1:      3     1
2:      9     4     1
3:     27    13     5    1
4:     81    40    18    6    1
5:    243   121    58   24    7    1
6:    729   364   179   82   31    8   1
7:   2187  1093   543  261  113   39   9   1
8:   6561  3280  1636  804  374  152  48  10  1
9:  19683  9841  4916 2440 1178  526 200  58 11  1
10: 59049 29524 14757 7356 3618 1704 726 258 69 12  1
... reformatted and extended. - _Wolfdieter Lang_, Jan 06 2015
----------------------------------------------------------
With the array M(k) as defined in the Formula section, the infinite product M(0)*M(1)*M(2)*... begins
/ 1        \/1           \/1        \       /1         \
| 3  1     ||0  1        ||0 1      |      | 3  1      |
| 9  4 1   ||0  3  1     ||0 0 1    |... = | 9  7  1   |
|27 13 5 1 ||0  9  4 1   ||0 0 3 1  |      |27 37 12 1 |
|...       ||0 27 13 5 1 ||0 0 9 4 1|      |...        |
|...       ||...         ||...      |      |...        |
= A143495. - _Peter Bala_, Dec 23 2014

Columns 1, 2, 3, 4, 5: A003462, A000340, A052150, A097786, A097787.

Cf. A055248, A143495.

a106516[n_] := Block[{a, k},
a[x_] := Flatten@ Last@ Reap[For[k = -1, k < x, Sow[Binomial[x, k] +
2 Sum[3^(i - 1)*Binomial[x - i, k], {i, 1, x}]], k++]]; Flatten@Array[a, n, 0]]; a106516[11] (* Michael De Vlieger, Dec 23 2014 *)

Riordan array (1/(1-3x), x/(1-x)); Number triangle T(n, 0)=A000244(n), T(n, k)=T(n-1, k-1)+T(n-1, k); T(n, k)=sum{j=0..n, binomial(n, k+j)2^j}.
From Peter Bala, Jul 16 2013: (Start)
T(n,k) = binomial(n,k) + 2*sum {i = 1..n} 3^(i-1)*binomial(n-i,k).
O.g.f.: (1 - t)/( (1 - 3*t)*(1 - (1 + x)*t) ) = 1 + (3 + x)*t + (9 + 4*x + x^2)*t^2 + ....
The n-th row polynomial R(n,x) = 1/(x - 2)*( x*(x + 1)^n - 2*3^n ). (End)
Closed-form formula for arbitrary left and right borders of Pascal-like triangle see A228196. - Boris Putievskiy, Aug 19 2013
T(n,k) = 4*T(n-1,k) + T(n-1,k-1) - 3*T(n-2,k) - 3*T(n-2,k-1), T(0,0)=1, T(1,0)=3, T(1,1)=1, T(n,k)=0 if k<0 or if k>n. - Philippe Deléham, Dec 26 2013
From Peter Bala, Dec 23 2014: (Start)
exp(x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(x)*(27 + 13*x + 5*x^2/2! + x^3/3!) = 27 + 40*x + 58*x^2/2! + 82*x^3/3! + 113*x^4/4! + .... The same property holds more generally for Riordan arrays of the form ( f(x), x/(1 - x) ).
Let M denote the present triangle. For k = 0,1,2,... define M(k) to be the lower unit triangular block array
/I_k 0\
\ 0  M/ having the k X k identity matrix I_k as the upper left block; in particular, M(0) = M. The infinite product M(0)*M(1)*M(2)*..., which is clearly well-defined, is equal to A143495 (but with a different offset). See the Example section. Cf. A055248. (End)
n-th row polynomial R(n, x) = (2*3^n - x*(1 + x)^n)/(2 - x). - Peter Bala, Mar 05 2025

A217764 Array defined by a(n,k) = floor((k+2)/2)3^n - floor((k+1)/2)2^n, read by antidiagonals.

Original entry on oeis.org

1, 3, 0, 9, 1, 1, 27, 5, 4, 0, 81, 19, 14, 2, 1, 243, 65, 46, 10, 5, 0, 729, 211, 146, 38, 19, 3, 1, 2187, 665, 454, 130, 65, 15, 6, 0, 6561, 2059, 1394, 422, 211, 57, 24, 4, 1, 19683, 6305, 4246, 1330, 665, 195, 84, 20, 7, 0, 59049, 19171, 12866, 4118, 2059, 633, 276, 76, 29, 5, 1

Offset: 0

Ross La Haye, Mar 23 2013

Columns 0,1,2,3 respectively correspond to relations R_3, R_4, R_0, R_1 defined in La Haye paper listed below.

			a(4,4) = 211 because floor((4+2)/2)*3^4 - floor((4+1)/2)*2^4 = 3*3^4 - 2*2^4 = 243 - 32 = 211.

Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.

Cf. a(1,k) = A084964(k+2); a(n,0) = A000244(n); a(n,1) = A001047(n); a(n,2) = A027649(n); a(n,3) = A056182(n); a(n,4) = A001047(n+1); a(n,5) = A210448(n); a(n,6) = A166060(n); a(n,7) = A145563(n); a(n,8) = A102485(n).

a(n,k) = floor((k+2)/2)*3^n - floor((k+1)/2)*2^n. a(n,k) = 5*a(n-1,k) - 6*a(n-2,k); a(0,k) = floor((k+2)/2) - floor((k+1)/2), a(1,k) = floor((k+2)/2)*3 - floor((k+1)/2)*2.

A002783 a(n) = 2*(3^n - 2^n) + 1.

Original entry on oeis.org

1, 3, 11, 39, 131, 423, 1331, 4119, 12611, 38343, 116051, 350199, 1054691, 3172263, 9533171, 28632279, 85962371, 258018183, 774316691, 2323474359, 6971471651, 20916512103, 62753730611, 188269580439, 564825518531, 1694510110023, 5083597438931, 15250926534519

Offset: 0

N. J. A. Sloane

Binomial transform of A095121. - R. J. Mathar, Oct 05 2012
Create a triangle having its left and right border both equal to the n-th row of Pascal's triangle, and internal terms m(i,j) = m(i-1,j-1) + m(i-1,j). Then the sum of all elements equals a(n). - J. M. Bergot, Oct 07 2012, edited by M. F. Hasler, Oct 10 2012
First differences of A090326 (with offset 1). - Wesley Ivan Hurt, Jul 08 2014

			From _J. M. Bergot_ and _M. F. Hasler_, Oct 10 2012: (Start)
For n=3, the triangle with left and right border (1,3,3,1) and internal terms m(i,j) = m(i-1,j-1) + m(i-1,j) is
     1
    3 3
   3 6 3
  1 9 9 1
and the sum of all the elements is 39 = a(3). (End)

H. Gupta, On a problem in parity, Indian J. Math., 11 (1969), 157-163.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
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 = 0..1000
H. Gupta, On a problem in parity, Indian J. Math., 11 (1969), 157-163. [Annotated scanned copy]
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Index entries for linear recurrences with constant coefficients, signature (6,-11,6).

Cf. A027649, A090326, A095121.

[2*(3^n - 2^n)+1 : n in [0..30]]; // Wesley Ivan Hurt, Jul 08 2014

A002783:=n->2*(3^n - 2^n)+1: seq(A002783(n), n=0..30); # Wesley Ivan Hurt, Jul 08 2014

CoefficientList[Series[(-1 + 3*x - 4*x^2)/((x - 1)*(3*x - 1)*(2*x - 1)), {x, 0, 30}], x] (* Wesley Ivan Hurt, Jul 08 2014 *)

a(n)=2*(3^n-2^n)+1 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: ( -1+3*x-4*x^2 ) / ( (x-1)*(3*x-1)*(2*x-1) ). - Simon Plouffe in his 1992 dissertation
a(n+1) - a(n) = 2*A027649(n). - R. J. Mathar, Oct 05 2012
E.g.f.: exp(x)*(1 - 2*exp(x) + 2*exp(2*x)). - Stefano Spezia, May 18 2024

More terms from Wesley Ivan Hurt, Jul 08 2014

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

A257286 a(n) = 56^n - 45^n.

Original entry on oeis.org

1, 10, 80, 580, 3980, 26380, 170780, 1087180, 6835580, 42575980, 263268380, 1618672780, 9907349180, 60420657580, 367406757980, 2228854610380, 13495197974780, 81581539411180, 492540994279580, 2970504754739980, 17899322473752380

Offset: 0

M. F. Hasler, May 03 2015

First differences of 6^n - 5^n = A005062.

a(n-1) is the number of numbers with n digits having the largest digit equal to 5. Or, equivalently, number of n-letter words over a 6-letter alphabet {a,b,c,d,e,f}, which must not start with the first letter of the alphabet, and in which the last letter of the alphabet must appear.

Index entries for linear recurrences with constant coefficients, signature (11,-30).

Cf. A005062.
Coincides with A125373 only for the first terms.
See also A000225, A027649, A255463, A257285, A257286, A257287, A257288, A257289 and A088924.

[5*6^n-4*5^n: n in [0..20]]; // Vincenzo Librandi, May 04 2015

Table[5 6^n - 4 5^n, {n, 0, 30}] (* Vincenzo Librandi, May 04 2015 *)

PARI
```
a(n)=5*6^n-4*5^n
```

a(n) = 11 a(n-1) - 30 a(n-2).
G.f.: (1-x)/((1-5*x)*(1-6*x)). - Vincenzo Librandi, May 04 2015
E.g.f.: exp(5*x)*(5*exp(x) - 4). - Stefano Spezia, Nov 15 2023

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.

A102285 G.f. (1-x)/(7x^2-6x+1).

Original entry on oeis.org

1, 5, 23, 103, 457, 2021, 8927, 39415, 174001, 768101, 3390599, 14966887, 66067129, 291634565, 1287337487, 5682582967, 25084135393, 110726731589, 488771441783, 2157541529575, 9523849084969, 42040303802789

Offset: 0

Creighton Dement, Feb 19 2005

A floretion-generated sequence relating to the second binomial transform of Pell numbers A000129.

Floretion Algebra Multiplication Program, FAMP Code: (a(n)) = jesforseq[ + .5'i + .5i' + 2'jj' + .5'ij' + .5'ji' ]; A000004 = vesforseq.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-7).

Cf. A086351, A027649, A007070 (inverse binomial transform), A081179, A163350 (binomial transform).

[Floor(((1+Sqrt(2))*(3+Sqrt(2))^n+(1-Sqrt(2))*(3-Sqrt(2))^n)/2): n in [0..30]]; // Vincenzo Librandi, Oct 12 2011

CoefficientList[Series[(1-x)/(7x^2-6x+1),{x,0,30}],x] (* or *) LinearRecurrence[{6,-7},{1,5},30] (* Harvey P. Dale, Dec 10 2017 *)

a(n) = A086351(n+1) - 3*A086351(n) (FAMP result); Inversion gives A027649 (SuperSeeker result); Inverse binomial transform of A007070 (SuperSeeker result);
From Al Hakanson (hawkuu(AT)gmail.com), Jul 25 2009: (Start)
a(n) = ((1+sqrt(2))*(3+sqrt(2))^n + (1-sqrt(2))*(3-sqrt(2))^n)/2 offset 0.
Third binomial transform of 1,2,2,4,4. (End)
a(n) = 6*a(n-1) - 7*a(n-2) for n > 1; a(0)=1, a(1)=5. - Philippe Deléham, Sep 19 2009
a(n) = A081179(n) + A086351(n). - Joseph M. Shunia, Sep 09 2019
a(n) = A081179(n+1)-A081179(n). - R. J. Mathar, Sep 11 2019

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 *)

cp's OEIS Frontend

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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A106516 A Pascal-like triangle based on 3^n.

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A217764 Array defined by a(n,k) = floor((k+2)/2)*3^n - floor((k+1)/2)*2^n, read by antidiagonals.

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A002783 a(n) = 2*(3^n - 2^n) + 1.

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

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A257286 a(n) = 5*6^n - 4*5^n.

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A217764 Array defined by a(n,k) = floor((k+2)/2)3^n - floor((k+1)/2)2^n, read by antidiagonals.

A257286 a(n) = 56^n - 45^n.

A102285 G.f. (1-x)/(7x^2-6x+1).