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Search: id:A141388
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| A141388 |
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Cubic form (k=3) of the generalized neo-combinations: t(n,m,k)=(n - m)^k*(m + 1)^k - 2^(n - 1). |
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1
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| 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
(list; table; graph; listen)
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OFFSET
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1,2
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COMMENT
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Row sums are:
{0, 12, 106, 528, 1923, 5696, 14540, 33152, 69141, 134096};
Note that the domain here is:
{m, 0, n - 1}], {n, 1, 10}
and not:
{m, 0, n }], {n, 0, 10}
The 2^(n-1) term was added to make the result generally symmetrical
on the Dynkin / A_n weight domain..
Triangular coefficient sequences of the general form:
t(n,m,k)=Floor[((n - m)^k*(m + 1)^k - 2^(n - 1))/n^k);
have a Pascal triangle / binomial shape.
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REFERENCES
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R. N. Cahn, Semi-Simple Lie Algebras and Their Representations, Dover, NY, 2006, ISBN 0-486-44999-8, p. 139.
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FORMULA
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k=3; t(n,m,k)=(n - m)^k*(m + 1)^k - 2^(n - 1).
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EXAMPLE
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{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}
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MATHEMATICA
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Clear[T, n, m, a]; T[n_, m_] = (n - m)^3*(m + 1)^3 - 2^(n - 1); a = Table[Table[T[n, m], {m, 0, n - 1}], {n, 1, 10}]; Flatten[a]
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CROSSREFS
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Cf. A003991.
Sequence in context: A004983 A034695 A053168 this_sequence A087236 A077193 A056482
Adjacent sequences: A141385 A141386 A141387 this_sequence A141389 A141390 A141391
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KEYWORD
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nonn,uned,tabl
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AUTHOR
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Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Aug 03 2008
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