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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.
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%I #61 Nov 27 2020 23:47:09

%S 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 1,28,71,103,109,86,43,1,1,36,97,151,177,166,121,57,1,1,45,127,208,

%U 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 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 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 T(n,k) = (k^2*(n-k)^2 + k*(k+2)*(n-k) + 2)/2 for 0 <= k <= n.

%F T(n,0) = T(n,n) = 1 for n >= 0.

%F 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 T(n,k) = T(n-1,k) + k * (1+k*(n-k)) for 0 <= k < n.

%F G.f. of column k >= 0: (1 + (k^2+k-2)*t + (1-k)*t^2) * t^k / (1-t)^3.

%F 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 The triangle T(n,k) for 0 <= k <= n starts:

%e n \k : 0 1 2 3 4 5 6 7 8 9 10 11 12

%e ====================================================================

%e 0 : 1

%e 1 : 1 1

%e 2 : 1 3 1

%e 3 : 1 6 7 1

%e 4 : 1 10 17 13 1

%e 5 : 1 15 31 34 21 1

%e 6 : 1 21 49 64 57 31 1

%e 7 : 1 28 71 103 109 86 43 1

%e 8 : 1 36 97 151 177 166 121 57 1

%e 9 : 1 45 127 208 261 271 235 162 73 1

%e 10 : 1 55 161 274 361 401 385 316 209 91 1

%e 11 : 1 66 199 349 477 556 571 519 409 262 111 1

%e 12 : 1 78 241 433 609 736 793 771 673 514 321 133 1

%e etc.

%t 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 (PARI) for(n=0,12,for(k=0,n,print1((k^2*(n-k)^2+k*(k+2)*(n-k)+2)/2,", "));print(" "))

%Y 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 nonn,easy,tabl

%O 0,5

%A _Werner Schulte_, Nov 26 2020