A141433 Triangle T(n, k) = (k-1)*(3*n-k), read by rows.

0, 0, 4, 0, 7, 12, 0, 10, 18, 24, 0, 13, 24, 33, 40, 0, 16, 30, 42, 52, 60, 0, 19, 36, 51, 64, 75, 84, 0, 22, 42, 60, 76, 90, 102, 112, 0, 25, 48, 69, 88, 105, 120, 133, 144, 0, 28, 54, 78, 100, 120, 138, 154, 168, 180

Offset

1,3

Links

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

Formula

G.f.: Sum_{k>=0} Sum_{n>=0} T(n,k)*x^n*y^k = y^2*x*(2*x+1-3*y)/((1-y)^3*(x-1)^2). (G.f. for the full array, not just the triangular subspace) - R. J. Mathar, Feb 19 2020

Sum_{k=1..n} T(n, k) = A304993(n-1) = (n-1)*n*(7*n -2)/6. - G. C. Greubel, Apr 01 2021

Example

Triangle begins as:
  0;
  0,  4;
  0,  7, 12;
  0, 10, 18, 24;
  0, 13, 24, 33,  40;
  0, 16, 30, 42,  52,  60;
  0, 19, 36, 51,  64,  75,  84;
  0, 22, 42, 60,  76,  90, 102, 112;
  0, 25, 48, 69,  88, 105, 120, 133, 144;
  0, 28, 54, 78, 100, 120, 138, 154, 168, 180;

Maple

A141433 := proc(n, m) (m-1)*(3*n-m) ; end proc:
seq(seq(A141433(n, m), m=1..n), n=1..18) ; # R. J. Mathar, Sep 14 2011

Mathematica

Flatten[Table[(m-1)(3n-m), {n, 10}, {m, n}]] (* Harvey P. Dale, Feb 04 2016 *)

Prog

(Magma) [(k-1)*(3*n-k): k in [1..n], n in [1..12]]; // G. C. Greubel, Apr 01 2021
(Sage) flatten([[(k-1)*(3*n-k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Apr 01 2021

Crossrefs

Cf. A304993 (row sums).

Sequence in context: A195286 A248931 A200501 * A019111 A182158 A103554

Adjacent sequences: A141430 A141431 A141432 * A141434 A141435 A141436

Keyword

nonn,easy,tabl

Author

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

Status

approved