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

0,8

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)

Links

Table of n, a(n) for n=0..81.

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)

Example

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

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 *)

Crossrefs

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

Generalization of A007318, A059259.

Sequence in context: A321748 A263447 A180303 * A292745 A047010 A047100

Adjacent sequences: A118920 A118921 A118922 * A118924 A118925 A118926

Keyword

easy,nonn,tabl

Author

Alford Arnold, May 05 2006

Extensions

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

Offset changed by Michael A. Allen, Nov 30 2021

Status

approved