Shuhei Tsujie - cp's OEIS frontend

A308440 Matrix product of triangle of Stirling numbers of second kind A008277 and square of unsigned Lah triangle A105278.

1, 5, 1, 37, 15, 1, 365, 223, 30, 1, 4501, 3675, 745, 50, 1, 66605, 68071, 18450, 1865, 75, 1, 1149877, 1411515, 479101, 64750, 3920, 105, 1, 22687565, 32512663, 13260030, 2244501, 181650, 7322, 140, 1, 503589781, 825175275, 393017185, 79948050, 8103711, 436590, 12558, 180, 1

Offset: 1

Shuhei Tsujie, May 27 2019

Also the number of k-dimensional flats of the extended Catalan arrangement of dimension n consisting of hyperplanes x_i - x_j = d (1 <= i < j <= n, -2 <= d <= 2).

			Triangle begins:
     1;
     5,    1;
    37,   15,   1;
   365,  223,  30,  1;
  4501, 3675, 745, 50, 1;
  ...

Robert Gill, The number of elements in a generalized partition semilattice, Discrete mathematics 186.1-3 (1998): 125-134.
N. Nakashima and S. Tsujie, Enumeration of Flats of the Extended Catalan and Shi Arrangements with Species, arXiv:1904.09748 [math.CO], 2019.

Cf. A008277, A105278, A050351 (first column), A109092 (row sums).

E.g.f.: exp((exp(x)-1)*y/(3-2exp(x))).

A308282 The fifth power of the unsigned Lah triangular matrix A105278.

Original entry on oeis.org

1, 10, 1, 150, 30, 1, 3000, 900, 60, 1, 75000, 30000, 3000, 100, 1, 2250000, 1125000, 150000, 7500, 150, 1, 78750000, 47250000, 7875000, 525000, 15750, 210, 1, 3150000000, 2205000000, 441000000, 36750000, 1470000, 29400, 280, 1, 141750000000, 113400000000, 26460000000, 2646000000, 132300000, 3528000, 50400, 360, 1

Offset: 1

Shuhei Tsujie, May 18 2019

Also the number of k-dimensional flats of the extended Shi arrangement of dimension n consisting of hyperplanes x_i - x_j = d (1 <= i < j <= n, -4 <= d <= 5).

			Triangle begins:
      1;
     10,     1;
    150,    30,    1;
   3000,   900,   60,   1;
  75000, 30000, 3000, 100, 1;
  ...

N. Nakashima and S. Tsujie, Enumeration of Flats of the Extended Catalan and Shi Arrangements with Species, arXiv:1904.09748 [math.CO], 2019.

Cf. A105278.

Table[5^(n - k) * Binomial[n - 1, k - 1] * n! / k!, {n, 1, 10}, {k, 1, n}] // Flatten (* Amiram Eldar, Jul 13 2019 *)

E.g.f.: exp(x*y/(1-5*x)).

T(n,k) = 5^(n-k)*binomial(n-1, k-1)*n!/k! = 5^(n-k)*A105278.

A308281 The third power of the unsigned Lah triangular matrix A105278.

Original entry on oeis.org

1, 6, 1, 54, 18, 1, 648, 324, 36, 1, 9720, 6480, 1080, 60, 1, 174960, 145800, 32400, 2700, 90, 1, 3674160, 3674160, 1020600, 113400, 5670, 126, 1, 88179840, 102876480, 34292160, 4762800, 317520, 10584, 168, 1, 2380855680, 3174474240, 1234517760, 205752960, 17146080, 762048, 18144, 216, 1

Offset: 1

Shuhei Tsujie, May 18 2019

Also the number of k-dimensional flats of the extended Shi arrangement of dimension n consisting of hyperplanes x_i - x_j = d (1 <= i < j <= n, -2 <= d <= 3).

			Triangle begins:
     1;
     6,    1;
    54,   18,    1;
   648,  324,   36,  1;
  9720, 6480, 1080, 60, 1;
  ...

N. Nakashima and S. Tsujie, Enumeration of Flats of the Extended Catalan and Shi Arrangements with Species, arXiv:1904.09748 [math.CO], 2019.

Cf. A105278.

Table[3^(n - k) * Binomial[n - 1, k - 1] * n! / k!, {n, 1, 10}, {k, 1, n}] // Flatten (* Amiram Eldar, Jul 13 2019 *)

E.g.f.: exp(x*y/(1-3*x)).

T(n,k) = 3^(n-k)*binomial(n-1, k-1)*n!/k! = 3^(n-k)*A105278.

cp's OEIS Frontend

User: Shuhei Tsujie

A308440 Matrix product of triangle of Stirling numbers of second kind A008277 and square of unsigned Lah triangle A105278.

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A308282 The fifth power of the unsigned Lah triangular matrix A105278.

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A308281 The third power of the unsigned Lah triangular matrix A105278.

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