A102365 Triangle T(n,k), 0 <= k <= n, read by rows: given by [ 1, 0, 3, 0, 5, 0, 7, 0, 9, 0, ...] DELTA [ 0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, ...] where DELTA is the operator defined in A084938.

1, 1, 0, 1, 1, 0, 1, 5, 1, 0, 1, 18, 15, 1, 0, 1, 58, 129, 37, 1, 0, 1, 179, 877, 646, 83, 1, 0, 1, 543, 5280, 8030, 2685, 177, 1, 0, 1, 1636, 29658, 82610, 56285, 10002, 367, 1, 0, 1, 4916, 159742, 756218, 919615, 335162, 34777, 749, 1, 0

Offset: 0

Author

Philippe Deléham, Feb 22 2005

Keywords

• nonn
• easy
• tabl

Comments

Generalized Eulerian numbers A008292.

Reversal of A211399. - Philippe Deléham, Feb 12 2013

Examples

			Triangle begins:
  1;
  1,  0;
  1,  1,   0;
  1,  5,   1,  0;
  1, 18,  15,  1, 0;
  1, 58, 129, 37, 1, 0; ...
		

Links

• G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Crossrefs

Diagonals: A000007, A000012, A050488, A142965, A142966.

Columns: A000012, A000340, A156922, A156923, A156924.

Programs

Mathematica

T[0, 0] := 1;  T[n_, -1] := 0;  T[n_, n_] := 0; T[n_, k_] := T[n, k] = (n - k)*T[n - 1, k - 1] + (2*k + 1)*T[n - 1, k]; Join[{1}, Table[If[k < 0, 0, If[k >= n, 0, T[n, k]]], {n, 1, 5}, {k, 0, n}] // Flatten] (* G. C. Greubel, Jun 30 2017 *)

Formula

T(n, k) = (n-k)*T(n-1, k-1) + (2*k+1)*T(n-1, k) with T(0, 0) = 1, T(0, k) = 0 if k > 0, T(n, k) = 0 if k < 0.

Sum_{k>=0} T(n, k)*2^k = A001147(n).

Sum_{k>=0} T(n, k) = A014307(n). - Philippe Deléham, Mar 19 2005