A374708 Triangle T read by rows: T(n,k) = (n - k)*n*(4*n^2 - 4*n*k + 2*k^2 - 1 + (-1)^k)/4, with 0 <= k < n.
1, 16, 4, 81, 36, 15, 256, 144, 80, 32, 625, 400, 255, 140, 65, 1296, 900, 624, 396, 240, 108, 2401, 1764, 1295, 896, 609, 364, 175, 4096, 3136, 2400, 1760, 1280, 864, 544, 256, 6561, 5184, 4095, 3132, 2385, 1728, 1215, 756, 369, 10000, 8100, 6560, 5180, 4080, 3100, 2320, 1620, 1040, 500
Offset: 1
Examples
n\k| 0 1 2 3 4 5
---+------------------------------
1 | 1
2 | 16 4
3 | 81 36 15
4 | 256 144 80 32
5 | 625 400 255 140 65
6 | 1296 900 624 396 240 108
...
For n = 3 the matrix M is
[ 1, 4, 15]
[ 4, 15, 32]
[15, 32, 65]
and therefore T(3, 0) = 1 + 15 + 65 = 81, T(3, 1) = 4 + 32 = 36, and T(3, 2) = 15.
Crossrefs
Programs
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Mathematica
T[n_,k_]:=(n-k)*n*(4*n^2 - 4*n*k+2*k^2-1+(-1)^k)/4; Table[T[n,k],{n,10},{k,0,n-1}]//Flatten
Formula
O.g.f.: x*(1 - 4*x^8*y^5 + x*(11 + 2*y) - x^7*y^4*(7 + 16*y) - x^2*(-11 + 6*y - 6*y^2) - x^5*y^2*(2 - 46*y - 3*y^2) - x^6*y^3*(-2 - 27*y + 4*y^2) - x^3*(-1 + 18*y + 38*y^2 - 2*y^3) - x^4*y*(2 + 14*y + 2*y^2 - y^3))/((1 - x)^5*(1 - x*y)^4*(1 + x*y)^2).
T(n,2) = A123865(n-1) for n > 1.
Comments