A209599 Triangle T(n,k), read by rows, given by (2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

1, 2, 0, 3, 1, 0, 5, 3, 0, 0, 8, 7, 1, 0, 0, 13, 15, 4, 0, 0, 0, 21, 30, 12, 1, 0, 0, 0, 34, 58, 31, 5, 0, 0, 0, 0, 55, 109, 73, 18, 1, 0, 0, 0, 0, 89, 201, 162, 54, 6, 0, 0, 0, 0, 0, 144, 365, 344, 145, 25, 1, 0, 0, 0, 0, 0

Offset: 0

Author

Philippe Deléham, Mar 10 2012

Keywords

• easy
• nonn
• tabl

Comments

A skew version of A122075.

Examples

			Triangle begins :
  1
  2, 0
  3, 1, 0
  5, 3, 0, 0
  8, 7, 1, 0, 0
  13, 15, 4, 0, 0, 0
  21, 30, 12, 1, 0, 0, 0
  34, 58, 31, 5, 0, 0, 0, 0
  55, 109, 73, 18, 1, 0, 0, 0, 0
  89, 201, 162, 54, 6, 0, 0, 0, 0, 0
  144, 365, 344, 145, 25, 1, 0, 0, 0, 0, 0
  ...
		

Links

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

Crossrefs

Cf. A122075, A122950, A000045, A023610, A129707

Programs

Mathematica

T[0, 0] := 1; T[1, 0] := 2; T[1, 1] := 0; T[n_, k_] := T[n, k] = If[n<0, 0, If[k > n, 0, T[n - 1, k] + T[n - 2, k] + T[n - 2, k - 1]]]; Table[T[n, k], {n, 0, 49}, {k, 0, n}] // Flatten (* G. C. Greubel, Dec 19 2017 *)

Formula

G.f.: (1+x)/(1-x-(1+y)*x^2).

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

Sum_{k, 0<=k<=n} T(n,k)*x^k = A040000(n), A000045(n+2), A000079(n), A006138(n), A026597(n), A133407(n), A133467(n), A133469(n), A133479(n), A133558(n), A133577(n), A063092(n) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 respectively.