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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.

%I A059098 #32 Dec 06 2023 14:18:46
%S A059098 1,1,1,2,3,2,5,10,12,6,15,37,62,60,24,52,151,320,450,360,120,203,674,
%T A059098 1712,3120,3720,2520,720,877,3263,9604,21336,33600,34440,20160,5040,
%U A059098 4140,17007,56674,147756,287784,394800,352800,181440,40320,21147,94828
%N 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.
%C A059098 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 A059098 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 A059098 E.g.f. for T(n, k): (exp(x)-1)^k*(exp(exp(x)-1)).
%F A059098 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 A059098 T(n, k) = k!*A049020(n, k). - _R. J. Mathar_, May 17 2016
%F A059098 T(n, k) = Sum_{j=0..k} (-1)^(k-j)*A046716(k, k-j)*Bell(n + j). - _Peter Luschny_, Dec 06 2023
%e A059098 Triangle begins:
%e A059098   [0] [ 1]
%e A059098   [1] [ 1,    1]
%e A059098   [2] [ 2,    3,    2]
%e A059098   [3] [ 5,   10,   12,    6]
%e A059098   [4] [15,   37,   62,   60,   24]
%e A059098   [5] [52,  151,  320,  450,  360,  120]
%e A059098   [6] [203, 674, 1712, 3120, 3720, 2520, 720]
%e A059098   ...;
%e A059098 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 A059098 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 A059098 T := proc(n, k) option remember; `if`(k < 0 or k > n, 0,
%p A059098       `if`(n = 0, 1, k*T(n-1, k-1) + (k+1)*T(n-1, k) + T(n-1, k+1)))
%p A059098     end:
%p A059098 seq(print(seq(T(n, k), k = 0..n)), n = 0..15); # _Peter Bala_, Oct 15 2023
%Y A059098 Cf. A000110(n) = T(n,0), A005493(n) = T(n,1), A059099 (row sums).
%Y A059098 Cf. A049020, A001861, A046716.
%K A059098 easy,nonn,tabl
%O A059098 0,4
%A A059098 _Vladeta Jovovic_, Jan 02 2001