A141431 Triangle T(n,k) = (k-1)*(3*n-k+1), read by rows.

0, 0, 5, 0, 8, 14, 0, 11, 20, 27, 0, 14, 26, 36, 44, 0, 17, 32, 45, 56, 65, 0, 20, 38, 54, 68, 80, 90, 0, 23, 44, 63, 80, 95, 108, 119, 0, 26, 50, 72, 92, 110, 126, 140, 152, 0, 29, 56, 81, 104, 125, 144, 161, 176, 189, 0, 32, 62, 90, 116, 140, 162, 182, 200, 216, 230, 0, 35, 68, 99, 128, 155, 180, 203, 224, 243, 260, 275

Offset

1,3

Links

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Formula

G.f.: Sum_{n>=0} Sum_{k>=0} T(n,k)*x^n*y^k = y^2*x*(x*y-4*y+x+2)/((1-y)^3*(1-x)^2). - R. J. Mathar, Nov 27 2015. x and y swapped to align with standard, 19 Feb 2020

Sum_{k=1..n} T(n, k) = (n-1)*n*(7*n+1)/6 = A245301(n-1). - G. C. Greubel, Mar 31 2021

Example

Triangle begins as:
  0;
  0,  5;
  0,  8, 14;
  0, 11, 20, 27;
  0, 14, 26, 36,  44;
  0, 17, 32, 45,  56,  65;
  0, 20, 38, 54,  68,  80,  90;
  0, 23, 44, 63,  80,  95, 108, 119;
  0, 26, 50, 72,  92, 110, 126, 140, 152;
  0, 29, 56, 81, 104, 125, 144, 161, 176, 189;

Maple

A141431 := proc(n, k)
        (k-1)*(3*n-k+1) ;
end proc:
seq(seq(A141431(n, k), k=1..n), n=1..14) ; # R. J. Mathar, Nov 10 2011

Mathematica

Table[(k-1)*(3*n-k+1), {n, 15}, {k, n}]//Flatten (* G. C. Greubel, Mar 31 2021 *)

Prog

(Magma) [(k-1)*(3*n-k+1): k in [1..n], n in [1..15]]; // G. C. Greubel, Mar 31 2021
(Sage) flatten([[(k-1)*(3*n-k+1) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Mar 31 2021

Crossrefs

Columns: A016789 (k=2), A016933 (k=3), A008591 (k=4).

Cf. A245301 (row sums).

Sequence in context: A199729 A240358 A200422 * A166011 A344144 A372364

Adjacent sequences: A141428 A141429 A141430 * A141432 A141433 A141434

Keyword

nonn,easy,tabl

Author

Roger L. Bagula and Gary W. Adamson, Aug 06 2008

Extensions

More terms added by G. C. Greubel, Mar 31 2021

Status

approved