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.

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

Offset

0,5

Comments

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).

Links

Table of n, a(n) for n=0..77.

Formula

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

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

T(n,k) = T(n-1,k-1) + k*(n-k)*(n-k+1) - (n-k)*(n-k-1)/2 for 0 < k <= n.

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

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

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

Example

The triangle T(n,k) for 0 <= k <= n starts:
n \k :  0   1    2    3    4    5    6    7    8    9   10   11   12
====================================================================
   0 :  1
   1 :  1   1
   2 :  1   3    1
   3 :  1   6    7    1
   4 :  1  10   17   13    1
   5 :  1  15   31   34   21    1
   6 :  1  21   49   64   57   31    1
   7 :  1  28   71  103  109   86   43    1
   8 :  1  36   97  151  177  166  121   57    1
   9 :  1  45  127  208  261  271  235  162   73    1
  10 :  1  55  161  274  361  401  385  316  209   91    1
  11 :  1  66  199  349  477  556  571  519  409  262  111    1
  12 :  1  78  241  433  609  736  793  771  673  514  321  133    1
etc.

Mathematica

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 *)

Prog

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

Crossrefs

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).

Sequence in context: A182823 A210866 A245474 * A339231 A133713 A008278

Adjacent sequences: A338366 A338367 A338368 * A338370 A338371 A338372

Keyword

nonn,easy,tabl

Author

Werner Schulte, Nov 26 2020

Status

approved