A360657 Number triangle T associated with 2-Stirling numbers and Lehmer-Comtet numbers (see Comments and Formula section).
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, 205, 20, 1, 0, 117649, 70993, 19981, 3410, 380, 27, 1, 0, 2097152, 1273609, 365001, 64701, 7770, 644, 35, 1, 0, 43046721, 26269505, 7628545, 1388310, 174951, 15834, 1022, 44, 1
Offset: 0
Examples
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)
Links
- Wikipedia, LU decomposition
- Aimin Xu, Determinants Involving the Numbers of the Stirling-Type, Filomat 33:6 (2019), 1659-1666.
Crossrefs
Programs
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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] ) ) ); }
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.
Comments