A336633 Triangle read by rows: T(n,k) is the number of generalized permutations related to the degenerate Eulerian numbers with exactly k ascents (0 <= k <= max(0,n-1)).
1, 1, 2, 2, 6, 16, 6, 24, 116, 116, 24, 120, 888, 1624, 888, 120, 720, 7416, 20984, 20984, 7416, 720, 5040, 67968, 270432, 419680, 270432, 67968, 5040, 40320, 682272, 3587904, 7861664, 7861664, 3587904, 682272, 40320, 362880, 7467840, 49701024, 144570624, 204403264, 144570624
Offset: 0
Examples
Triangle T(n,k) (with rows n >= 0 and columns k = 0..max(0,n-1)) begins:
1;
1;
2, 2;
6, 16, 6;
24, 116, 116, 24;
120, 888, 1624, 888, 120;
720, 7416, 20984, 20984, 7416, 720;
5040, 67968, 270432, 419680, 270432, 67968, 5040;
...
Links
- Orli Herscovici, Generalized permutations related to the degenerate Eulerian numbers, arXiv preprint arXiv:2007.13205 [math.CO], 2020.
Programs
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Maple
Tnk[0, 0] := 1; for n to N do for k from 0 to n do if 0 < k and k < n then Tnk[n, k] := (n + k)*Tnk[n - 1, k] + (2*n - k - 1)*Tnk[n - 1, k - 1]; else if k = 0 then Tnk[n, k] := (n + k)*Tnk[n - 1, k]; else Tnk[n, k] := 0; end if; end if; end do; end do
Formula
T(n,k) = (n+k)*T(n-1,k) + (2*n-k-1)*T(n-1,k-1) for positive integers n and 0 <= k < n; T(0,0)=1 (or T(1,0)=1); otherwise T(n,k)=0.
From Peter Bala, Jan 08 2021: (Start)
The following remarks are all conjectures:
The e.g.f. (without the initial 1) A(x,t) = x + (2 + 2*t)*x^2/2! + (6 + 16*t + 6*t^2)*x^3/3! + ... satisfies the autonomous differential equation dA/dx = (1 + A)^2*(1 + t*A)^2.
The series reversion of A(x,t) with respect to x equals Integral_{u = 0..x} 1/((1 + u)^2*(1 + t*u)^2) du.
Let f(x,t) = (1 + x)^2*(1 + t*x)^2 and let D be the operator f(x,t)*d/dx. Then the (n+1)-th row polynomial = D^n(f(x,t)) evaluated at x = 0. (End)