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A141388
Triangle T(n, k) = ( k*(n-k+1) )^3 - 2^(n-1), read by rows.
1
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.
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
KEYWORD
sign,tabl,look
AUTHOR
Roger L. Bagula, Aug 03 2008
EXTENSIONS
Edited by G. C. Greubel, Apr 01 2021
STATUS
approved