A118923 Triangle T(n,k) built by placing T(n,0)=A000012(n) in the left edge, T(n,n)=A079978(n) on the right edge and filling the body with the Pascal recurrence T(n,k) = T(n-1,k) + T(n-1,k-1).

1, 1, 0, 1, 1, 0, 1, 2, 1, 1, 1, 3, 3, 2, 0, 1, 4, 6, 5, 2, 0, 1, 5, 10, 11, 7, 2, 1, 1, 6, 15, 21, 18, 9, 3, 0, 1, 7, 21, 36, 39, 27, 12, 3, 0, 1, 8, 28, 57, 75, 66, 39, 15, 3, 1, 1, 9, 36, 85, 132, 141, 105, 54, 18, 4, 0, 1, 10, 45, 121, 217, 273, 246, 159, 72, 22, 4, 0, 1, 11, 55, 166

Offset: 0

Author

Alford Arnold, May 05 2006

Keywords

• easy
• nonn
• tabl

Comments

The fourth diagonal is 1, 2, 5, 11, 21, ..., which is 1 + A000292. The fifth diagonal is 0, 2, 7, 18, 39, 75, 132, 217, 338, 504, 725, 1012, ..., which is A051743.

The array A007318 is generated by placing A000012 on both edges with the same Pascal-like recurrence, and the array A059259 uses edges defined by A000012 and A059841. - R. J. Mathar, Jan 21 2008

From Michael A. Allen, Nov 30 2021: (Start)

T(n,n-k) is the (n,k)-th entry of the (1/(1-x^3), x/(1-x)) Riordan array.

Sums of rows give A077947.

Sums of antidiagonals give A079962. (End)

Examples

			The table begins
  1
  1  0
  1  1  0
  1  2  1  1
  1  3  3  2  0
  1  4  6  5  2  0
  1  5 10 11  7  2  1
  1  6 15 21 18  9  3  0
		

Crossrefs

Cf. A000292, A079978, A008620, A079998, A051743, A077947.

Generalization of A007318, A059259.

Programs

Maple

A000012 := proc(n) 1 ; end: A079978 := proc(n) if n mod 3 = 0 then 1; else 0 ; fi ; end: A118923 := proc(n,k) if k = 0 then A000012(n); elif k = n then A079978(n) ; else A118923(n-1,k)+A118923(n-1,k-1) ; fi ; end: for n from 0 to 15 do for k from 0 to n do printf("%d, ",A118923(n,k)) ; od: od: # R. J. Mathar, Jan 21 2008

Mathematica

Flatten@Table[CoefficientList[Series[1/((1 + x*y + x^2*y^2)(1 - x - x*y)), {x, 0, 23}, {y, 0, 11}], {x, y}][[n + 1, k + 1]], {n, 0, 11}, {k, 0, n}] (* Michael A. Allen, Nov 30 2021 *)

Formula

From Michael A. Allen, Nov 30 2021: (Start)

For 0 <= k < n, T(n,k) = (n-k)*Sum_{j=0..floor(k/3)} binomial(n-3*j,n-k)/(n-3*j).

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

Extensions

Edited and extended by R. J. Mathar, Jan 21 2008

Offset changed by Michael A. Allen, Nov 30 2021