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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.
2

%I #32 Dec 06 2023 14:18:46

%S 1,1,1,2,3,2,5,10,12,6,15,37,62,60,24,52,151,320,450,360,120,203,674,

%T 1712,3120,3720,2520,720,877,3263,9604,21336,33600,34440,20160,5040,

%U 4140,17007,56674,147756,287784,394800,352800,181440,40320,21147,94828

%N 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.

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

%H Marc Chamberland, <a href="https://doi.org/10.1016/j.laa.2011.08.030">Factored matrices can generate combinatorial identities</a>, Linear Algebra and its Applications, Volume 438, Issue 4, 2013, pp. 1667-1677.

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

%F 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

%F T(n, k) = k!*A049020(n, k). - _R. J. Mathar_, May 17 2016

%F T(n, k) = Sum_{j=0..k} (-1)^(k-j)*A046716(k, k-j)*Bell(n + j). - _Peter Luschny_, Dec 06 2023

%e Triangle begins:

%e [0] [ 1]

%e [1] [ 1, 1]

%e [2] [ 2, 3, 2]

%e [3] [ 5, 10, 12, 6]

%e [4] [15, 37, 62, 60, 24]

%e [5] [52, 151, 320, 450, 360, 120]

%e [6] [203, 674, 1712, 3120, 3720, 2520, 720]

%e ...;

%e 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 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 + ...

%p T := proc(n, k) option remember; `if`(k < 0 or k > n, 0,

%p `if`(n = 0, 1, k*T(n-1, k-1) + (k+1)*T(n-1, k) + T(n-1, k+1)))

%p end:

%p seq(print(seq(T(n, k), k = 0..n)), n = 0..15); # _Peter Bala_, Oct 15 2023

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

%Y Cf. A049020, A001861, A046716.

%K easy,nonn,tabl

%O 0,4

%A _Vladeta Jovovic_, Jan 02 2001