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

0,6

Comments

Row sums: A133494.

Links

Table of n, a(n) for n=0..54.

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

Example

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)

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 *)

Crossrefs

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

Sequence in context: A021498 A025251 A193334 * A011119 A240747 A120553

Adjacent sequences: A185078 A185079 A185080 * A185082 A185083 A185084

Keyword

easy,nonn,tabl

Author

Philippe Deléham, Jan 22 2012

Extensions

Corrected by Jean-François Alcover, Jun 20 2017

Status

approved