[seqfan] Related to certain partitions of n^2, n^3, n^4, n^5, ...

[Ed Jeffery]

Hello,

We have, for N=0,1,..., that

(N+1)^3 = N^3+3*N^2+3*N+1
        = (N+1)*(N+2)*(N+3)/6 + N*(N+1)*(5*N+7)/6,

so, for n=1,2,..., it follows that

n^3 = n*(n+1)*(n+2)/6 + (n-1)*n*(5*n+2)/6

    = A000292(n)+A162148(n-1)

    = A000578(n)

(see [1,2 and 3]). For n,k=1,2,..., the above relation (and there are
several others) is immediately seen in the following aerated triangle T, in
which one considers sums of the products T(n,k)*k for which the summation
is split in an obvious way to produce the correct two sequences above.

.n\k.1.2.3.4.5.6.7.8.9.10.11.12.13.14.15.16.17.18.19.20.21.22...
.1...1;
.2...0.2.0.1;
.3...1.0.3.0.2.0.1;
.4...0.2.0.4.0.3.0.2.0..1;
.5...1.0.3.0.5.0.4.0.3..0..2..0..1;
.6...0.2.0.4.0.6.0.5.0..4..0..3..0..2..0..1;
.7...1.0.3.0.5.0.7.0.6..0..5..0..4..0..3..0..2..0..1;
.8...0.2.0.4.0.6.0.8.0..7..0..6..0..5..0..4..0..3..0..2..0..1;
...

I think that this triangle is not in OEIS. Row n has 3*n-2 entries. We have
Sum_{k=1,...,3*n-2} T(n,k)*k = n^3, and the row sums are
{1,3,7,12,19,27,37,48,...} which seems to be A077043 [4] up to an offset.

See also triangle A176850 [5] (for the n^4 case) which I found
independently while working on this idea years ago and which can be derived
from the above triangle. In fact the following triangle A was derived from
A176850 and is not in OEIS (and from which similar relations for n^5 can be
found).

n\k..1..2..3..4..5..6..7..8..9.10.11.12.13.14.15.16.17.18.19.20.21...
1....1;
2....0..5..0..4..0..1;
3....6..0.15..0.15..0.10..0..4..0..1;
4....0.20..0.34..0.36..0.30..0.20..0.10..0..4..0..1;
5...16..0.45..0.65..0.70..0.64..0.51..0.35..0.20..0.10..0..4..0..1;
...

Row n has 5*n-4 entries, and Sum_{k=1,...,5*n-4} A(n,k)*k = n^5. Row sums
are {1,10,51,155,381,780,1451,...} which seems to be sequence A077044 [6]
up to an offset.  The diagonal starting at 1,1 is
{1,5,15,34,65,111,175,260,369,505,...} which seems to be A006003 [7] up to
an offset. I counted something like thirty or forty cross references
related to triangles T and A, and I am sure there are many more. Finally,
my original motivation for this was my own naive consideration of Fermat's
last theorem (before the proof by Andrew Wiles).

LEJ

References:

[1] A000292, https://oeis.org/A000292
[2] A000578, https://oeis.org/A000578
[3] A162148, https://oeis.org/A162148
[4] A077043, https://oeis.org/A077043
[5] A176850, https://oeis.org/A176850 <https://oeis.org/A176850>
[6] A077044, https://oeis.org/A077044
[7] A006003, https://oeis.org/A006003