A080416 Stirling-like number triangle defined by paired decomposition of C(n+3,3) = A000292.

1, 1, 1, 1, 4, 1, 1, 12, 10, 1, 1, 32, 67, 20, 1, 1, 80, 376, 252, 35, 1, 1, 192, 1909, 2560, 742, 56, 1, 1, 448, 9094, 22928, 12346, 1848, 84, 1, 1, 1024, 41479, 189120, 177599, 46912, 4074, 120, 1, 1, 2304, 183412, 1472704, 2318149

Offset: 0

Author

Paul Barry, Feb 17 2003

Keywords

• easy
• nonn
• tabl

Comments

Note that the Stirling numbers of the second kind are generated in a similar fashion by decomposing the triangular numbers C(n+2,2) as {1}, {1,2}, {1,2,3}, .... The defining sequence A000292 appears as the subdiagonal when the triangle is arranged in lower-triangular form. The second column is A001787.

Gives the number of ways to construct pairs of k-block partitions from 1 to n such that the sum of the minima of the i-th block of the first partition and the (k-i+1)th block of the second partition is n+1. - Ken Joffaniel M Gonzales, Jun 13 2010

Examples

			Rows are
  {1},
  {1,  1},
  {1,  4,  1},
  {1, 12, 10,  1},
  {1, 32, 67, 20,  1},
  ...
		

Links

• R. B. Corcino, K. J. M. Gonzales, M. J. C. Loquias and E. L. Tan, Dually weighted Stirling-type sequences, arXiv preprint arXiv:1302.4694 [math.CO], 2013-2014.
• R. B. Corcino, K. J. M. Gonzales, M. J. C. Loquias and E. L. Tan, Dually weighted Stirling-type sequences, Europ. J. Combin., 43, 2015, 55-67.

Crossrefs

Cf. A000292, A001787, A008277, A080251.

Programs

Mathematica

s[b_, n_, k_] := s[b, n, k] = Which[n==k==0, 1, n==0, 0, k==0, 0, True, s[b+1, n-1, k-1] + k*b*s[b, n-1, k]]
Table[s[0, n+2, k+2], {n, 0, 10}, {k, 0, n}] // Flatten
(* a specialization of equation (9) in the Corcino et al. paper *)
(* Mikhail Lavrov, Oct 12 2022 *)
T[ n_, k_] := If[n < 0, 0, SeriesCoefficient[x^k / Product[1 + x*(Floor[j/2] + 1)*(Floor[j/2] - k - 1), {j,0,k}], {x,0,n}]]; (* Michael Somos, Oct 12 2022 *)

PARI

{T(n, k) = if(n<0, 0, polcoeff(x^k / prod(j=0, k, 1 + x*(j\2 + 1)*(j\2 - k - 1) + x*O(x^n)), n))}; /* Michael Somos, Oct 12 2022 */

Formula

Columns are generated as follows: Display C(n+3, 3) as row sums of the triangle A080251, or {1}, {2, 2}, {3, 3, 4}, {4, 4, 6, 6}, {5, 5, 8, 8, 9}, ... The columns are then generated by 1/(1-x), 1/(1-2x)^2, 1/((1-3x)^2*(1-4x)), 1/((1-4x)^2*(1-6x)^2), etc.