OFFSET
0,8
LINKS
Shishuo Fu, Z. Lin, J. Zeng, Two new unimodal descent polynomials, arXiv preprint arXiv:1507.05184 [math.CO], 2015-2019.
FORMULA
The g.f. F(x,y) = Sum_{n>=2,1<=k<=n-1} T(n,k)x^n/n!y^k satisfies the partial differential equation (1-xy) D_{x}F + (y^2-y) D_{y}F = F + 1 - e^(-xy). (Is there a closed form solution?)
T(n,k) = (k+1) * T(n-1,k) + (n-k) * T(n-1,k-1) + (-1)^n * delta(k,n-1), where delta(,) is the Kronecker delta.
EXAMPLE
Triangle begins:
n\k | 0 1 2 3 4 5 6 7
----+--------------------------------
0 | 1;
1 | 0, 0;
2 | 0, 1 0;
3 | 0, 2, 0 0;
4 | 0, 4, 4, 1 0;
5 | 0, 8, 24, 12, 0 0;
6 | 0, 16, 104, 120, 24, 1 0;
7 | 0, 32, 392, 896, 480, 54, 0, 0;
T(4,2) = 4 counts 2143, 3142, 3421, 4312.
MATHEMATICA
u[n_, 0] := 0; u[n_, k_] /; k == n-1 := If [EvenQ[n], 1, 0]; u[n_, k_] /; 1 <= k <= n - 2 := (n - k) u[n - 1, k - 1] + (k + 1) u[n - 1, k]; Table[u[n, k], {n, 2, 10}, {k, n - 1}]
PROG
(PARI) T(n, k) = if(n==0, k==0, (k+1)*T(n-1, k)+(n-k)*T(n-1, k-1)+(-1)^n*(k==n-1)); \\ Seiichi Manyama, Apr 25 2026
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
David Callan, Nov 29 2012
EXTENSIONS
Edited by Seiichi Manyama, Apr 25 2026
STATUS
approved
