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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).
3
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)
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
Generalization of A007318, A059259.
Sequence in context: A321748 A263447 A180303 * A292745 A047010 A047100
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