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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).

%I A198065 #12 Mar 20 2023 03:44:15
%S A198065 0,1,1,32,6,32,243,63,63,243,1024,364,192,364,1024,3125,1365,665,665,
%T A198065 1365,3125,7776,3906,2016,1458,2016,3906,7776,16807,9331,5187,3367,
%U A198065 3367,5187,9331,16807,32768,19608,11648,7448,6144,7448,11648,19608,32768,59049
%N 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).
%F A198065 T(n,k) = 3*n*k^4-6*k^3*n^2+7*k^2*n^3-4*k*n^4+n^5.
%F A198065 T(n,0) = T(n,n) = n^m = n^5 = A000584(n).
%F A198065 T(2n,n) = (m+1)n^m = 6n^5.
%F A198065 T(2n+1,n+1) = (n+1)^(m+1)-n^(m+1) = (n+1)^6-n^6 = A022522(n).
%F A198065 Sum{k=0..n} T(n,k) = (13n^6+30n^5+20n^4-3n^2)/30.
%F A198065 T(n+1,k+1)C(n,k)^6/(k+1)^5 = A197655(n,k).
%e A198065 [0]                        0
%e A198065 [1]                       1, 1
%e A198065 [2]                    32, 6, 32
%e A198065 [3]                 243, 63, 63, 243
%e A198065 [4]            1024, 364, 192, 364, 1024
%e A198065 [5]         3125, 1365, 665, 665, 1365, 3125
%e A198065 [6]     7776, 3906, 2016, 1458, 2016, 3906, 7776
%e A198065 [7] 16807, 9331, 5187, 3367, 3367, 5187, 9331, 16807
%p A198065 A198065 := (n,k) -> 3*n*k^4-6*k^3*n^2+7*k^2*n^3-4*k*n^4+n^5:
%o A198065 (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
%Y A198065 Cf. A057427, A003056, A073254, A198063, A198064.
%Y A198065 Cf. A000584, A022522, A197655.
%K A198065 nonn,tabl
%O A198065 0,4
%A A198065 _Peter Luschny_, Oct 26 2011