Power Sums via CSN
My native language is not English, so I hope AI helps me make myself understood.
Brief summary about CSN in sums of powers:
From Knuth, we know that:
n^4 = 12 * S(n+2, 4) + (S(n+1, 2) = n^2)
where S(n, k) = C(n-1, k) + C(n, k).
The approach to express "any" sum of powers using the basis {C, S, N (Narayana numbers)} only requires elementary arithmetic and recognizing sequences.
For example, let's say we want to compute the sum of fifth powers (k^5):
We just multiply the sequences S(n+2, 4) and n^2 by n, which obviously gives us n^3.
We compute the terms of S(n+2, 4) = 0, 1, 6, 20, 50, ...
So: 0*1 = 0, 1*2 = 2, 6*3 = 18, 20*4 = 80, 50*5 = 250. Divide by 2 and we get 0, 1, 9, 40, 125. If we sum this sequence cumulatively, we get 0, 1, 10, 50, 175, which matches the first 5 terms of N(n+2, 4).
Therefore, the formula for SIGMA(k^5) = 24 * N(n+2, 4) + C(n+1, 2)^2.
IMPORTANT SECTION
I can predict all the initial coefficients (this is a conjecture), but once they reappear, it's no longer that easy.
This is where the concept of a "memory matrix" comes in: an array that contains all the coefficients that couple to the initial coefficients, making it possible to express any power sum.
In this matrix, I've noticed some interesting things:
The only non-trivial pattern in any diagonal or column appears in the second diagonal: the sequence 2, 10, 28, 60, 110, ... which is twice S(n+2, 3) = 1, 5, 14, 30, 55, ...
We can also observe an uneven growth pattern in the diagonals, where some terms grow faster than others. For example, in the second diagonal, the terms (2, 0, 0, -3, -11, 13/2) are always smaller than their two nearest terms, that is, compared to 1, 2, 10, 28, 60, 110.
Therefore, we can conjecture that, when taking absolute values, one subsequence of the diagonal shows slower growth compared to the other subsequence. This could be useful for generating coefficients by establishing upper or lower bounds.
Although there is no formula that generates all the coefficients of the matrix, its columns show the following asymptotic behavior in the limit as p goes to infinity:
Ratio (delta_11 / delta_9):
0 / 1 = 0
33 / 10 = 3.3
20 / 10 = 2.0
5733 / 1216 ≈ 4.714
15960 / 3419 ≈ 4.668
6473741 / 1377146 ≈ 4.700
70977060 / 15098484 ≈ 4.701
29913085305 / 6360805200 ≈ 4.702
Ratio (delta_11 / delta_13):
10 / 1 = 10
10 / 0 = undefined
1216 / 294 ≈ 4.136
3419 / 828 ≈ 4.129
1377146 / 349923 ≈ 3.936
15098484 / 3836868 ≈ 3.935
6360805200 / 1626520060 ≈ 3.910
So we have:
· The ratio column(delta_9)/column(delta_11) converges to 4.7
· The ratio column(delta_11)/column(delta_13) converges to 3.9
· The ratio column(delta_13)/column(delta_15) converges to 13.6
· The ratio column(delta_15)/column(delta_17) converges to 15.1
Thus, we can assume that the ratio between any pair of consecutive columns converges to a constant value.
Does this seem interesting to you?