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A157636 Triangle read by rows: T(n, k) = 1 if k=0 or k=n, otherwise = n*k*(n-k)/2.

%I A157636 #10 Sep 08 2022 08:45:42
%S A157636 1,1,1,1,1,1,1,3,3,1,1,6,8,6,1,1,10,15,15,10,1,1,15,24,27,24,15,1,1,
%T A157636 21,35,42,42,35,21,1,1,28,48,60,64,60,48,28,1,1,36,63,81,90,90,81,63,
%U A157636 36,1,1,45,80,105,120,125,120,105,80,45,1
%N A157636 Triangle read by rows: T(n, k) = 1 if k=0 or k=n, otherwise = n*k*(n-k)/2.
%H A157636 G. C. Greubel, <a href="/A157636/b157636.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A157636 T(n, k) = 1 if k=0 or k=n, otherwise = n*k*(n-k)/2.
%F A157636 Sum_{k=0..n} T(n, k) = 2 + n^2*(n^2 - 1)/12 = 2 + A002415(n) if n>0.
%F A157636 From _G. C. Greubel_, Dec 13 2021: (Start)
%F A157636 T(n, k) = T(n, n-k).
%F A157636 T(n, 1) = [n<2] + binomial(n, 2).
%F A157636 T(n, 2) = A132411(n-1), for n >= 2.
%F A157636 T(2*n, n) = [n=0] + A000578(n). (End)
%e A157636 Triangle begins as:
%e A157636   1;
%e A157636   1,  1;
%e A157636   1,  1,  1;
%e A157636   1,  3,  3,   1;
%e A157636   1,  6,  8,   6,   1;
%e A157636   1, 10, 15,  15,  10,   1;
%e A157636   1, 15, 24,  27,  24,  15,   1;
%e A157636   1, 21, 35,  42,  42,  35,  21,   1;
%e A157636   1, 28, 48,  60,  64,  60,  48,  28,  1;
%e A157636   1, 36, 63,  81,  90,  90,  81,  63, 36,  1;
%e A157636   1, 45, 80, 105, 120, 125, 120, 105, 80, 45, 1;
%t A157636 T[n_, k_] = If[n*k*(n-k)==0, 1, n*k*(n-k)/2];
%t A157636 Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten
%o A157636 (Magma) A157636:= func< n,k | k eq 0 or k eq n select 1 else n*k*(n-k)/2 >;
%o A157636 [A157636(n,k): k in [0..n], n in [0..15]]; // _G. C. Greubel_, Dec 13 2021
%o A157636 (Sage)
%o A157636 def A157636(n,k): return 1 if (k==0 or k==n) else n*k*(n-k)/2
%o A157636 flatten([[A157636(n,k) for k in (0..n)] for n in (0..15)]) # _G. C. Greubel_, Dec 13 2021
%Y A157636 Cf. A000578, A002415, A107985, A132411.
%K A157636 nonn,tabl,easy
%O A157636 0,8
%A A157636 _Roger L. Bagula_, Mar 03 2009