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A141388 Cubic form (k=3) of the generalized neo-combinations: t(n,m,k)=(n - m)^k*(m + 1)^k - 2^(n - 1). 0
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; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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.

REFERENCES

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

LINKS

Table of n, a(n) for n=1..49.

FORMULA

k=3; t(n,m,k)=(n - m)^k*(m + 1)^k - 2^(n - 1).

EXAMPLE

{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}

MATHEMATICA

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]

CROSSREFS

Cf. A003991.

Sequence in context: A325998 A322216 A053168 * A255473 A255295 A255475

Adjacent sequences:  A141385 A141386 A141387 * A141389 A141390 A141391

KEYWORD

nonn,uned,tabl

AUTHOR

Roger L. Bagula, Aug 03 2008

STATUS

approved

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Last modified September 25 20:01 EDT 2020. Contains 337344 sequences. (Running on oeis4.)