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

0,3

Links

Table of n, a(n) for n=0..42.

Orli Herscovici, Generalized permutations related to the degenerate Eulerian numbers, arXiv preprint arXiv:2007.13205 [math.CO], 2020.

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)

Example

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;
  ...

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

Crossrefs

Columns k = 0..1 give: A000142, A288964. Row sums give A007559.

Cf. A008292, A008517.

Sequence in context: A260687 A083555 A233147 * A001464 A067136 A180068

Adjacent sequences: A336630 A336631 A336632 * A336634 A336635 A336636

Keyword

easy,nonn,tabf

Author

Orli Herscovici, Jul 28 2020

Status

approved