OFFSET
0,5
COMMENTS
Triangle T is created using 2-Stirling numbers of the first (A049444) and the second (A143494) kind. The unusual construction is as follows:
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.
Matrix inverse of T is A360753. - Werner Schulte, Feb 21 2023
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
LINKS
Wikipedia, LU decomposition
Aimin Xu, Determinants Involving the Numbers of the Stirling-Type, Filomat 33:6 (2019), 1659-1666.
FORMULA
For the definition of triangle T see Comments section.
Conjectured formulas:
1. T(n, k) = (Sum_{i=k..n} A354794(n, i) * (i-1)!) / (k-1)! for 0 < k <= n.
2. T(n, k) - k * T(n, k+1) = A354794(n, k) for 0 <= k <= n.
3. T(n, 1) = A000169(n) = n^(n-1) for n > 0.
4. T(n, 2) = A055869(n-1) = n^(n-1) - (n-1)^(n-1) for n > 1.
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.
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.
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.
EXAMPLE
Triangle T(n, k), 0 <= k <= n, starts:
n\k : 0 1 2 3 4 5 6 7 8 9
==========================================================================
0 : 1
1 : 0 1
2 : 0 2 1
3 : 0 9 5 1
4 : 0 64 37 9 1
5 : 0 625 369 97 14 1
6 : 0 7776 4651 1275 205 20 1
7 : 0 117649 70993 19981 3410 380 27 1
8 : 0 2097152 1273609 365001 64701 7770 644 35 1
9 : 0 43046721 26269505 7628545 1388310 174951 15834 1022 44 1
etc.
From Peter Bala, Oct 10 2023: (Start)
LU factorization of the square array of Stirling numbers of the second kind (apply Xu, Lemma 2.2):
/ 1 \ / 1 1 1 1 ...\ / 1 1 1 1 ... \
| 1 1 || 2 5 9 ...| | 1 3 6 10 ... |
| 1 3 1 || 9 37 ...| = | 1 7 25 65 ... |
| 1 7 6 1 || 64 ...| | 1 15 90 350 ... |
| ... || ...| | ... |
(End)
PROG
(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] ) ) ); }
CROSSREFS
KEYWORD
AUTHOR
Werner Schulte, Feb 15 2023
STATUS
approved