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A157634 Triangle T(n, k) = 1 if k = 0 or k = n, otherwise n^5 - k^5 - (n-k)^5, read by rows.

%I A157634 #13 Sep 08 2022 08:45:42
%S A157634 1,1,1,1,30,1,1,210,210,1,1,780,960,780,1,1,2100,2850,2850,2100,1,1,
%T A157634 4650,6720,7290,6720,4650,1,1,9030,13650,15540,15540,13650,9030,1,1,
%U A157634 15960,24960,29400,30720,29400,24960,15960,1,1,26280,42210,51030,54900,54900,51030,42210,26280,1
%N A157634 Triangle T(n, k) = 1 if k = 0 or k = n, otherwise n^5 - k^5 - (n-k)^5, read by rows.
%H A157634 G. C. Greubel, <a href="/A157634/b157634.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A157634 T(n, k) = 1 if k = 0 or k = n, otherwise 5*n*k*(n-k)*(n^2 -n*k +k^2).
%F A157634 T(n, n-k) = T(n, k).
%F A157634 Sum_{k=0..n} T(n, k) = 2 - [n=0] + 30*A006858(n).
%F A157634 From _G. C. Greubel_, Dec 13 2021: (Start)
%F A157634 T(n, 1) = [n<2] + 30*A006325(n).
%F A157634 T(2*n, n) = [n=0] + 30*A000584(n). (End)
%e A157634 Triangle begins as:
%e A157634   1;
%e A157634   1,     1;
%e A157634   1,    30,     1;
%e A157634   1,   210,   210,     1;
%e A157634   1,   780,   960,   780,     1;
%e A157634   1,  2100,  2850,  2850,  2100,     1;
%e A157634   1,  4650,  6720,  7290,  6720,  4650,     1;
%e A157634   1,  9030, 13650, 15540, 15540, 13650,  9030,     1;
%e A157634   1, 15960, 24960, 29400, 30720, 29400, 24960, 15960,     1;
%e A157634   1, 26280, 42210, 51030, 54900, 54900, 51030, 42210, 26280,     1;
%e A157634   1, 40950, 67200, 82950, 91200, 93750, 91200, 82950, 67200, 40950, 1;
%t A157634 T[n_, k_]:= If[n*k*(n-k)==0, 1, n^5 - (k^5 + (n-k)^5)];
%t A157634 Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten
%o A157634 (Magma)
%o A157634 A157634:= func< n,k | k eq 0 or k eq n select 1 else n^5 - (k^5 + (n-k)^5) >;
%o A157634 [A157634(n, k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Dec 13 2021
%o A157634 (Sage)
%o A157634 def A157634(n,k): return 1 if (k==0 or k==n) else n^5 - (k^5 + (n-k)^5)
%o A157634 flatten([[A157634(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Dec 13 2021
%Y A157634 Cf. A000584, A006325, A006858.
%K A157634 nonn,tabl,easy
%O A157634 0,5
%A A157634 _Roger L. Bagula_, Mar 03 2009