A198065 Triangle read by rows (n >= 0, 0 <= k <= n, m = 5); T(n,k) = Sum{j=0..m} Sum{i=0..m} (-1)^(j+i)*C(i,j)*n^j*k^(m-j).

0, 1, 1, 32, 6, 32, 243, 63, 63, 243, 1024, 364, 192, 364, 1024, 3125, 1365, 665, 665, 1365, 3125, 7776, 3906, 2016, 1458, 2016, 3906, 7776, 16807, 9331, 5187, 3367, 3367, 5187, 9331, 16807, 32768, 19608, 11648, 7448, 6144, 7448, 11648, 19608, 32768, 59049

Offset: 0

Author

Peter Luschny, Oct 26 2011

Keywords

• nonn
• tabl

Examples

			[0]                        0
[1]                       1, 1
[2]                    32, 6, 32
[3]                 243, 63, 63, 243
[4]            1024, 364, 192, 364, 1024
[5]         3125, 1365, 665, 665, 1365, 3125
[6]     7776, 3906, 2016, 1458, 2016, 3906, 7776
[7] 16807, 9331, 5187, 3367, 3367, 5187, 9331, 16807
		

Crossrefs

Cf. A057427, A003056, A073254, A198063, A198064.

Cf. A000584, A022522, A197655.

Programs

Magma

&cat[[n*(k^2-k*n+n^2)*(3*k^2-3*k*n+n^2): k in [0..n]]: n in [0..9]];  // Bruno Berselli, Nov 02 2011

Maple

A198065 := (n,k) -> 3*n*k^4-6*k^3*n^2+7*k^2*n^3-4*k*n^4+n^5:

Formula

T(n,k) = 3*n*k^4-6*k^3*n^2+7*k^2*n^3-4*k*n^4+n^5.

T(n,0) = T(n,n) = n^m = n^5 = A000584(n).

T(2n,n) = (m+1)n^m = 6n^5.

T(2n+1,n+1) = (n+1)^(m+1)-n^(m+1) = (n+1)^6-n^6 = A022522(n).

Sum{k=0..n} T(n,k) = (13n^6+30n^5+20n^4-3n^2)/30.

T(n+1,k+1)C(n,k)^6/(k+1)^5 = A197655(n,k).