This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A338369 #61 Nov 27 2020 23:47:09
%S A338369 1,1,1,1,3,1,1,6,7,1,1,10,17,13,1,1,15,31,34,21,1,1,21,49,64,57,31,1,
%T A338369 1,28,71,103,109,86,43,1,1,36,97,151,177,166,121,57,1,1,45,127,208,
%U A338369 261,271,235,162,73,1,1,55,161,274,361,401,385,316,209,91,1,1,66,199,349,477,556,571,519,409,262,111,1
%N A338369 Triangle read by rows: T(n,k) = (Sum_{i=0..n-k}(1+k*i)^3)/(Sum_{i=0..n-k} (1+k*i)) for 0 <= k <= n.
%C A338369 Seen as a square array: (1) A(n,k) = T(n+k,k) = (k^2*n^2+k*(k+2)*n+2)/2 for n,k >= 0; (2) A(n,k) = A(n-1,k) + k*(1 + k*n) for k >= 0 and n > 0; (3) A(n,k) = A(n,k-1) + k*n*(n+1) - n*(n-1)/2 for n >= 0 and k > 0; (4) G.f. of row n >= 0: (2 + (n^2+3*n-4)*x + (n^2-n+2)*x^2) / (2*(1-x)^3).
%F A338369 T(n,k) = (k^2*(n-k)^2 + k*(k+2)*(n-k) + 2)/2 for 0 <= k <= n.
%F A338369 T(n,0) = T(n,n) = 1 for n >= 0.
%F A338369 T(n,k) = T(n-1,k-1) + k*(n-k)*(n-k+1) - (n-k)*(n-k-1)/2 for 0 < k <= n.
%F A338369 T(n,k) = T(n-1,k) + k * (1+k*(n-k)) for 0 <= k < n.
%F A338369 G.f. of column k >= 0: (1 + (k^2+k-2)*t + (1-k)*t^2) * t^k / (1-t)^3.
%F A338369 E.g.f.: exp(x+y)*(2 + (x^2 + 2*x - 2)*y + (x^2 - 4*x + 2)*y^2 - (2*x - 5)*y^3 + y^4)/2. - _Stefano Spezia_, Nov 27 2020
%e A338369 The triangle T(n,k) for 0 <= k <= n starts:
%e A338369 n \k : 0 1 2 3 4 5 6 7 8 9 10 11 12
%e A338369 ====================================================================
%e A338369 0 : 1
%e A338369 1 : 1 1
%e A338369 2 : 1 3 1
%e A338369 3 : 1 6 7 1
%e A338369 4 : 1 10 17 13 1
%e A338369 5 : 1 15 31 34 21 1
%e A338369 6 : 1 21 49 64 57 31 1
%e A338369 7 : 1 28 71 103 109 86 43 1
%e A338369 8 : 1 36 97 151 177 166 121 57 1
%e A338369 9 : 1 45 127 208 261 271 235 162 73 1
%e A338369 10 : 1 55 161 274 361 401 385 316 209 91 1
%e A338369 11 : 1 66 199 349 477 556 571 519 409 262 111 1
%e A338369 12 : 1 78 241 433 609 736 793 771 673 514 321 133 1
%e A338369 etc.
%t A338369 T[n_, k_] := Sum[(1 + k*i)^3, {i, 0, n - k}]/Sum[1 + k*i, {i, 0, n - k}]; Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* _Amiram Eldar_, Nov 26 2020 *)
%o A338369 (PARI) for(n=0,12,for(k=0,n,print1((k^2*(n-k)^2+k*(k+2)*(n-k)+2)/2,", "));print(" "))
%Y A338369 Cf. A000012 (column 0, main diagonal), A000217 (column 1), A056220 (column 2), A081271 (column 3), A118057 (column 4), A002061 (1st subdiagonal), A056109 (2nd subdiagonal), A085473 (3rd subdiagonal), A272039 (4th subdiagonal).
%K A338369 nonn,easy,tabl
%O A338369 0,5
%A A338369 _Werner Schulte_, Nov 26 2020