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A360657 Number triangle T associated with 2-Stirling numbers and Lehmer-Comtet numbers (see Comments and Formula section).

%I A360657 #35 Dec 17 2023 11:22:07
%S A360657 1,0,1,0,2,1,0,9,5,1,0,64,37,9,1,0,625,369,97,14,1,0,7776,4651,1275,
%T A360657 205,20,1,0,117649,70993,19981,3410,380,27,1,0,2097152,1273609,365001,
%U A360657 64701,7770,644,35,1,0,43046721,26269505,7628545,1388310,174951,15834,1022,44,1
%N A360657 Number triangle T associated with 2-Stirling numbers and Lehmer-Comtet numbers (see Comments and Formula section).
%C A360657 Triangle T is created using 2-Stirling numbers of the first (A049444) and the second (A143494) kind. The unusual construction is as follows:
%C A360657   Define A(n, k) by recurrence A(n, k) = A(n-1, k-1) + (k+1) * A(n-1, k) for 0 < k < n with initial values A(n, n) = 1, n >= 0, and A(n, 0) = 0, n > 0. A without column k = 0 is A143494. Let B = A^(-1) matrix inverse of A. B without column k = 0 is A049444. Now define T(m, k) = Sum_{i=0..m-k} B(m-k, i) * A(m-1+i, m-1) for 0 < k <= m = n/2 and T(m, 0) = 0^m for 0 <= m = n/2; T(i, j) = 0 if i < j or j < 0.
%C A360657 Matrix inverse of T is A360753. - _Werner Schulte_, Feb 21 2023
%C A360657 Conjecture: the transpose of this array is the upper triangular matrix U in the LU factorization of the array of Stirling numbers of the second kind read as a square array; the corresponding lower triangular array L is the triangle of Stirling numbers of the second kind. See the example section below. - _Peter Bala_, Oct 10 2023
%H A360657 Wikipedia, <a href="https://en.wikipedia.org/wiki/LU_decomposition">LU decomposition</a>
%H A360657 Aimin Xu, <a href="https://doi.org/10.2298/FIL1906659X">Determinants Involving the Numbers of the Stirling-Type</a>, Filomat 33:6 (2019), 1659-1666.
%F A360657 For the definition of triangle T see Comments section.
%F A360657 Conjectured formulas:
%F A360657 1. T(n, k) = (Sum_{i=k..n} A354794(n, i) * (i-1)!) / (k-1)! for 0 < k <= n.
%F A360657 2. T(n, k) - k * T(n, k+1) = A354794(n, k) for 0 <= k <= n.
%F A360657 3. T(n, 1) = A000169(n) = n^(n-1) for n > 0.
%F A360657 4. T(n, 2) = A055869(n-1) = n^(n-1) - (n-1)^(n-1) for n > 1.
%F A360657 5. T(n, k) = (Sum_{i=0..k-1} (-1)^i * binomial(k-1, i) * (n-i)^(n-1)) / (k-1)! for 0 < k <= n.
%F A360657 6. Sum_{i=1..n} (-1)^(n-i) * binomial(n-1+k, i-1) * T(n, i) * (i-1)! = (k-1)^(n-1) for n > 0 and k >= 0.
%F A360657 7. Matrix product of A354795 and T without column 0 equals A094587.
%F A360657 8. Matrix product of T and A354795 without column 0 equals A088956.
%F A360657 9. E.g.f. of column k > 0: Sum_{n>=k} T(n, k) * t^(n-1) / (n-1)! = (W(-t)/(-t)) * (Sum_{n>=k} A354794(n, k) * t^(n-1) / (n-1)!) where W is the Lambert_W-function.
%e A360657 Triangle T(n, k), 0 <= k <= n, starts:
%e A360657 n\k :  0         1         2        3        4       5      6     7   8  9
%e A360657 ==========================================================================
%e A360657   0 :  1
%e A360657   1 :  0         1
%e A360657   2 :  0         2         1
%e A360657   3 :  0         9         5        1
%e A360657   4 :  0        64        37        9        1
%e A360657   5 :  0       625       369       97       14       1
%e A360657   6 :  0      7776      4651     1275      205      20      1
%e A360657   7 :  0    117649     70993    19981     3410     380     27     1
%e A360657   8 :  0   2097152   1273609   365001    64701    7770    644    35   1
%e A360657   9 :  0  43046721  26269505  7628545  1388310  174951  15834  1022  44  1
%e A360657   etc.
%e A360657 From _Peter Bala_, Oct 10 2023: (Start)
%e A360657 LU factorization of the square array of Stirling numbers of the second kind (apply Xu, Lemma 2.2):
%e A360657  / 1               \ / 1   1   1   1  ...\    / 1   1   1    1  ... \
%e A360657  | 1   1           ||      2   5   9  ...|   |  1   3   6   10  ... |
%e A360657  | 1   3   1       ||          9  37  ...| = |  1   7  25   65  ... |
%e A360657  | 1   7   6   1   ||             64  ...|   |  1  15  90  350  ... |
%e A360657  | ...             ||                 ...|   |  ...                 |
%e A360657 (End)
%o A360657 (PARI) tabl(m) = {my(n=2*m, A = matid(n), B, T); for( i = 2, n, for( j = 2, i, A[i, j] = A[i-1, j-1] + j * A[i-1, j] ) ); B = A^(-1); T = matrix( m, m, i, j, if( j == 1, 0^(i-1), sum( r = 0, i-j, B[i-j+1, r+1] * A[i-1+r, i-1] ) ) ); }
%Y A360657 Cf. A049444, A143494, A354794, A354795, A088956, A094587.
%Y A360657 Cf. A000007 (column 0), A000169 (column 1), A055869 (column 2).
%Y A360657 Cf. A000012 (main diagonal), A000096 (1st subdiagonal), A360753 (matrix inverse).
%K A360657 nonn,easy,tabl
%O A360657 0,5
%A A360657 _Werner Schulte_, Feb 15 2023