OFFSET
0,5
COMMENTS
The row polynomials of this triangle P(n, x) = Sum_{m=0..n} T(n, m)*x^m appear as numerator polynomials in the o.g.f.s for the diagonal sequences of triangle A132393 (|Stirling1| with offset 0 for rows and columns). See the comment and the P. Bala link there.
LINKS
Seiichi Manyama, Rows n = 0..139, flattened
Andrew Elvey Price, Alan D. Sokal, Phylogenetic trees, augmented perfect matchings, and a Thron-type continued fraction (T-fraction) for the Ward polynomials, arXiv:2001.01468 [math.CO], 2020.
Wikipedia, Eulerian numbers of the second kind
FORMULA
EXAMPLE
The triangle T(n, m) begins:
n\m 0 1 2 3 4 5 6 7 8 9 ...
0: 1
1: 0 1
2: 0 2 1
3: 0 6 8 1
4: 0 24 58 22 1
5: 0 120 444 328 52 1
6: 0 720 3708 4400 1452 114 1
7: 0 5040 33984 58140 32120 5610 240 1
8: 0 40320 341136 785304 644020 195800 19950 494 1
9: 0 362880 3733920 11026296 12440064 5765500 1062500 67260 1004 1
...
MAPLE
T:= (n, k)-> combinat[eulerian2](n, n-k):
seq(seq(T(n, k), k=0..n), n=0..12); # Alois P. Heinz, Jul 26 2017
MATHEMATICA
Table[Boole[n == 0] + Sum[(-1)^(n + k) * Binomial[2 n + 1, k] StirlingS1[2 n - m - k, n - m - k], {k, 0, n - m - 1}], {n, 0, 10}, {m, n, 0, -1}] // Flatten (* Michael De Vlieger, Jul 21 2017, after Jean-François Alcover at A201637 *)
CROSSREFS
T(2n,n) gives A290306.
KEYWORD
AUTHOR
Wolfdieter Lang, Jul 20 2017
STATUS
approved