A185081 Triangle T(n,k), read by rows, given by (0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

1, 0, 1, 0, 1, 2, 0, 2, 4, 3, 0, 3, 9, 10, 5, 0, 5, 18, 28, 22, 8, 0, 8, 35, 68, 74, 45, 13, 0, 13, 66, 154, 210, 177, 88, 21, 0, 21, 122, 331, 541, 574, 397, 167, 34, 0, 34, 222, 686, 1302, 1656, 1446, 850, 310, 55

Offset: 0

Author

Philippe Deléham, Jan 22 2012

Keywords

• easy
• nonn
• tabl

Comments

Row sums: A133494.

Examples

			Triangle begins:
  1;
  0,  1;
  0,  1,  2;
  0,  2,  4,  3;
  0,  3,  9, 10,  5;
  0,  5, 18, 28, 22,  8;
  0,  8, 35, 68, 74, 45, 13;
From _Philippe Deléham_, Apr 11 2012: (Start)
Triangle in A209138 begins:
  1;
  1,  2;
  2,  4,  3;
  3,  9, 10,  5;
  5, 18, 28, 22,  8;
  8, 35, 68, 74, 45, 13; (End)
		

Crossrefs

Cf. A000007, A000045, A000244, A004798, A033999, A084938, A133494, A209138.

Programs

Mathematica

nmax = 9; T[n_, n_] := Fibonacci[n+1]; T[, 0] = 0; T[n, 1] := Fibonacci[n]; T[n_, k_] /; 1 < k < n := T[n, k] = T[n - 1, k] + T[n - 1, k - 1] + T[n - 2, k] + T[n - 2, k - 1] + T[n - 2, k - 2]; T[, ] = 0;
Table[T[n, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 20 2017 *)

Formula

Sum_{k=0..n} T(n,k)*x^k = A033999(n), A000007(n), A133494(n) for x = -1, 0, 1 respectively.

T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k) + T(n-2,k-1) + T(n-2,k-2), for n > 2, T(0,0) = T(1,1) = T(2,1) = 1, T(1,0) = T(2,0) = 0, T(2,2) = 2.

T(n+1,n) = A004798(n), T(n,n) = T(n+1,1) = A000045(n+1).

T(n,k) = A209138(n,k-1) for k >= 1. - Philippe Deléham, Apr 11 2012

G.f.: (-1 + x^2*y + x + x^2)/(-1 + x^2*y + x + x^2 + x*y + x^2*y^2). - R. J. Mathar, Aug 11 2015

Extensions

Corrected by Jean-François Alcover, Jun 20 2017