A141388 Triangle T(n, k) = ( k*(n-k+1) )^3 - 2^(n-1), read by rows.

0, 6, 6, 23, 60, 23, 56, 208, 208, 56, 109, 496, 713, 496, 109, 184, 968, 1696, 1696, 968, 184, 279, 1664, 3311, 4032, 3311, 1664, 279, 384, 2616, 5704, 7872, 7872, 5704, 2616, 384, 473, 3840, 9005, 13568, 15369, 13568, 9005, 3840, 473, 488, 5320, 13312, 21440, 26488, 26488, 21440, 13312, 5320, 488

Offset

1,2

Comments

From row n = 23 onward every term of this triangle is negative. - G. C. Greubel, Apr 01 2021

References

R. N. Cahn, Semi-Simple Lie Algebras and Their Representations, Dover, NY, 2006, ISBN 0-486-44999-8, p. 139.

Links

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

Formula

T(n, k) = ( k*(n-k+1) )^3 - 2^(n-1).

Sum_{k=1..n} T(n, K) = (1/70)*binomial(n+2,3)*(16 +18*n +21*n^2 +12*n^3 +3*n^4) - 2^(n-1)*n. - G. C. Greubel, Apr 01 2021

Example

Triangle begins as:
    0;
    6,    6;
   23,   60,    23;
   56,  208,   208,    56;
  109,  496,   713,   496,   109;
  184,  968,  1696,  1696,   968,   184;
  279, 1664,  3311,  4032,  3311,  1664,   279;
  384, 2616,  5704,  7872,  7872,  5704,  2616,   384;
  473, 3840,  9005, 13568, 15369, 13568,  9005,  3840,  473;
  488, 5320, 13312, 21440, 26488, 26488, 21440, 13312, 5320, 488;

Maple

A141388:= (n, k)-> ( k*(n-k+1) )^3 - 2^(n-1); seq(seq(A141388(n, k), k=1..n), n=1..12); # G. C. Greubel, Apr 01 2021

Mathematica

Table[(n-k+1)^3*k^3 - 2^(n-1), {n, 10}, {k, n}]//Flatten

Prog

(Magma) [( k*(n-k+1) )^3 - 2^(n-1): k in [1..n], n in [1..12]]; // G. C. Greubel, Apr 01 2021
(Sage) flatten([[( k*(n-k+1) )^3 - 2^(n-1) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Apr 01 2021

Crossrefs

Sequence in context: A325998 A322216 A053168 * A255473 A255295 A255475

Adjacent sequences: A141385 A141386 A141387 * A141389 A141390 A141391

Keyword

sign,tabl,look

Author

Roger L. Bagula, Aug 03 2008

Extensions

Edited by G. C. Greubel, Apr 01 2021

Status

approved