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A059098
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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.
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2
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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
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OFFSET
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0,4
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COMMENTS
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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
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LINKS
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FORMULA
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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
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EXAMPLE
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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 + ...
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MAPLE
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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
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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