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A198063 Triangle read by rows (n >= 0, 0 <= k <= n, m = 3); T(n,k) = Sum{j=0..m} Sum{i=0..m} (-1)^(j+i)*C(i,j)*n^j*k^(m-j).

%I A198063 #17 Sep 08 2022 08:45:59
%S A198063 0,1,1,8,4,8,27,15,15,27,64,40,32,40,64,125,85,65,65,85,125,216,156,
%T A198063 120,108,120,156,216,343,259,203,175,175,203,259,343,512,400,320,272,
%U A198063 256,272,320,400,512,729,585,477,405,369,369,405,477,585,729
%N A198063 Triangle read by rows (n >= 0, 0 <= k <= n, m = 3); T(n,k) = Sum{j=0..m} Sum{i=0..m} (-1)^(j+i)*C(i,j)*n^j*k^(m-j).
%C A198063 Read as an infinite symmetric square array, this is the table A(n,k)=(n+k)(n^2+k^2), cf. A321500 for the triangle with k <= n. - _M. F. Hasler_, Nov 22 2018
%H A198063 G. C. Greubel, <a href="/A198063/b198063.txt">Rows n=0..100 of triangle, flattened</a>
%F A198063 T(n,k) = 2*k^2*n - 2*k*n^2 + n^3.
%F A198063 T(n,0) = T(n,n) = n^m = n^3 = A000578(n).
%F A198063 T(2*n,n) = (m+1)n^m = 4*n^3 = A033430(n).
%F A198063 T(2*n+1,n+1) = (n+1)^(m+1) - n^(m+1) = (n+1)^4 - n^4 = A005917(n).
%F A198063 Sum{k=0..n} T(n,k) = (2*n^4 + 3*n^3 + n^2)/3 = A098077(n).
%F A198063 T(n+1,k+1)*C(n,k)^4/(k+1)^3 = A197653(n,k).
%e A198063 [0]                   0
%e A198063 [1]                  1, 1
%e A198063 [2]                8, 4, 8
%e A198063 [3]             27, 15, 15, 27
%e A198063 [4]           64, 40, 32, 40, 64
%e A198063 [5]        125, 85, 65, 65, 85, 125
%e A198063 [6]   216, 156, 120, 108, 120, 156, 216
%e A198063 [7] 343, 259, 203, 175, 175, 203, 259, 343
%e A198063 From _M. F. Hasler_, Nov 22 2018: (Start)
%e A198063 Can also be seen as the square array A(n,k)=(n+k)*(n^2 + k^2) read by antidiagonals:
%e A198063 n | k: 0   1   2   3 ...
%e A198063 --+----------------------
%e A198063 0 |    0   1   8  27 ...
%e A198063 1 |    1   4  15  40 ...
%e A198063 2 |    8  15  32  65 ...
%e A198063 3 |   27  40  65 108 ...
%e A198063 ...      ...     ...
%e A198063 (End)
%p A198063 A198063 := (n,k) -> 2*k^2*n-2*k*n^2+n^3:
%t A198063 t[n_, k_] := 2 k^2*n - 2 k*n^2 + n^3; Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Amiram Eldar_, Nov 22 2018 *)
%o A198063 (PARI) A198063(n,k)=2*k^2*n-2*k*n^2+n^3 \\ See also A321500. - _M. F. Hasler_, Nov 22 2018
%o A198063 (Magma) [[2*k^2*n-2*k*n^2+n^3: k in [0..n]]: n in [0..12]]; // _G. C. Greubel_, Nov 23 2018
%o A198063 (Sage) [[ 2*k^2*n-2*k*n^2+n^3 for k in range(n+1)] for n in range(12)] # _G. C. Greubel_, Nov 23 2018
%Y A198063 Cf. A057427, A003056, A073254, A198064, A198065.
%K A198063 nonn,tabl
%O A198063 0,4
%A A198063 _Peter Luschny_, Oct 26 2011