A143198 Triangle read by rows T(n, k) = n + (n+1)*(k-1)*(k+2)/2.

-1, -1, 1, -1, 2, 8, -1, 3, 11, 23, -1, 4, 14, 29, 49, -1, 5, 17, 35, 59, 89, -1, 6, 20, 41, 69, 104, 146, -1, 7, 23, 47, 79, 119, 167, 223, -1, 8, 26, 53, 89, 134, 188, 251, 323, -1, 9, 29, 59, 99, 149, 209, 279, 359, 449, -1, 10, 32, 65, 109, 164, 230, 307, 395, 494, 604

Offset

0,5

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Formula

T(n, k) = (n+1)*A000096(k-1) + n, for 0 <= k <= n, n >= 0.

Sum_{k=0..n} T(n, k) = (n-1)*(n+1)*(n^2+4*n+6)/6 (row sums).

Sum_{k=0..n} (-1)^k*T(n, k) = ((-1)^n*(2*n^3+6*n^2+5*n-3) -n-5)/8. - G. C. Greubel, Jun 09 2024

Example

Triangle begins as:
  -1;
  -1,  1;
  -1,  2,  8;
  -1,  3, 11, 23;
  -1,  4, 14, 29,  49;
  -1,  5, 17, 35,  59,  89;
  -1,  6, 20, 41,  69, 104, 146;
  -1,  7, 23, 47,  79, 119, 167, 223;
  -1,  8, 26, 53,  89, 134, 188, 251, 323;
  -1,  9, 29, 59,  99, 149, 209, 279, 359, 449;
  -1, 10, 32, 65, 109, 164, 230, 307, 395, 494, 604;

Maple

A143198 := proc(n, m)
        n+(n+1)*(m-1)*(m+2)/2 ;
end proc:
seq(seq(A143198(n, k), k=0..n), n=0..12) ; # R. J. Mathar, Nov 11 2011

Mathematica

Table[n+(n+1)*(k-1)*(k+2)/2, {n, 0, 15}, {k, 0, n}]//Flatten

Prog

(Magma) [n+(n+1)*(k-1)*(k+2)/2: k in [0..n], n in [0..15]]; // G. C. Greubel, Jun 09 2024
(SageMath) flatten([[n+(n+1)*(k-1)*(k+2)/2 for k in range(n+1)] for n in range(16)]) # G. C. Greubel, Jun 09 2024

Crossrefs

Cf. A000096.

Sequence in context: A242070 A155694 A200689 * A100064 A352589 A275980

Adjacent sequences: A143195 A143196 A143197 * A143199 A143200 A143201

Keyword

easy,sign,tabl

Author

Roger L. Bagula and Gary W. Adamson, Oct 20 2008

Status

approved