A059098 Triangle read by rows. T(n, k) = Sum_{i=0..n} Stirling2(n, i)*Product_{j=1..k} (i - j + 1) for 0 <= k <= n.

1, 1, 1, 2, 3, 2, 5, 10, 12, 6, 15, 37, 62, 60, 24, 52, 151, 320, 450, 360, 120, 203, 674, 1712, 3120, 3720, 2520, 720, 877, 3263, 9604, 21336, 33600, 34440, 20160, 5040, 4140, 17007, 56674, 147756, 287784, 394800, 352800, 181440, 40320, 21147, 94828

Offset: 0

Author

Vladeta Jovovic, Jan 02 2001

Keywords

• easy
• nonn
• tabl

Comments

The transpose of this lower unitriangular array is the U factor in the LU decomposition of the Hankel matrix (Bell(i+j-2))A000110(n).%20The%20L%20factor%20is%20A049020%20(see%20Chamberland,%20p.%201672).%20-%20_Peter%20Bala">i,j >= 1, where Bell(n) = A000110(n). The L factor is A049020 (see Chamberland, p. 1672). - _Peter Bala, Oct 15 2023

Examples

			Triangle begins:
  [0] [ 1]
  [1] [ 1,    1]
  [2] [ 2,    3,    2]
  [3] [ 5,   10,   12,    6]
  [4] [15,   37,   62,   60,   24]
  [5] [52,  151,  320,  450,  360,  120]
  [6] [203, 674, 1712, 3120, 3720, 2520, 720]
  ...;
E.g.f. for T(n, 2) = (exp(x)-1)^2*(exp(exp(x)-1)) = x^2 + 2*x^3 + 31/12*x^4 + 8/3*x^5 + 107/45*x^6 + 343/180*x^7 + 28337/20160*x^8 + 349/360*x^9 + ...;
E.g.f. for T(n, 3) = (exp(x)-1)^3*(exp(exp(x)-1)) = x^3 + 5/2*x^4 + 15/4*x^5 + 13/3*x^6 + 127/30*x^7 + 1759/480*x^8 + 34961/12096*x^9 + ...
		

Links

• Marc Chamberland, Factored matrices can generate combinatorial identities, Linear Algebra and its Applications, Volume 438, Issue 4, 2013, pp. 1667-1677.

Crossrefs

Cf. A000110(n) = T(n,0), A005493(n) = T(n,1), A059099 (row sums).

Cf. A049020, A001861, A046716.

Programs

Maple

T := proc(n, k) option remember; `if`(k < 0 or k > n, 0,
      `if`(n = 0, 1, k*T(n-1, k-1) + (k+1)*T(n-1, k) + T(n-1, k+1)))
    end:
seq(print(seq(T(n, k), k = 0..n)), n = 0..15); # Peter Bala, Oct 15 2023

Formula

E.g.f. for T(n, k): (exp(x)-1)^k*(exp(exp(x)-1)).

n-th row is M^n*[1,0,0,0,...], where M is a tridiagonal matrix with all 1's in the superdiagonal, (1,2,3,...) in the main and subdiagonals; and the rest zeros. - Gary W. Adamson, Jun 23 2011

T(n, k) = k!*A049020(n, k). - R. J. Mathar, May 17 2016

T(n, k) = Sum_{j=0..k} (-1)^(k-j)*A046716(k, k-j)*Bell(n + j). - Peter Luschny, Dec 06 2023