[seqfan] A posting to the Sequence Fans list from George Beck

Neil Sloane njasloane at gmail.com
Sun Apr 24 16:10:49 CEST 2016

(George Beck (george.beck at gmail.com) asked me to post this on his behalf.
He defines some interesting transformations of sequences.  I have edited
the message slightly. )
Dear Seqfans,
Rereading an old email from Olivier Gerard, I was startled to see that the
g.f. of the sigma function was the logarithmic derivative of the g.f. of
partition function.  I began wondering again about the pentagonal number
theorem, which applies to both the partition function p A000041 and the sum
of divisors sigma_1 A147843. Why aren't there more such formulas?

In looking at some proofs, I saw the definition sigma(0) = n when
calculating sigma(n). How can sigma(0) know n?

Shevelev (arxiv.org/pdf/0903.1743) called the difference between f(n) and
the sum f(n-1) + f(n-2) - f(n-5) - f(n-12) + ... the compensating sequence
for f.

Thinking of that sum as a sequence transform, define MM(f)(n) = f(n) -
f(n-1) - f(n-2) + f(n-5) + ..., with signs and places given by A010815, the
signed characteristic function of the generalized pentagonal numbers
A001318. The MM stands for MacMahon.

In those terms, the pentagonal number theorem for partitions is MM(p) = 0.
For the sum of the divisors, it is MM(sigma) = -n A010815(n).

Malefant (http://arxiv.org/abs/1103.1585) expresses q(n) (odd partitions,
A000009) in terms of p(n) in equation (34):
q(n) = p(n) - p(n-2) - p(n-4) + p(n-10) + ... +(-1)^m p(n - 2 p_m)+ ... ,
where the p_m are A001318. Then MM(q) = sgn(2 p_m), A152749.

Substituting q for p in the previous sum, let
w(m) = q(n) - q(n-2) - q(n-4) + q(n-10) + ... + (-1)^m q(n - 2 p_m).
Then w(n) is the characteristic function of the triangular numbers A000217,
but MM(w) is irregular.

A036820 contains this comment from Omar E. Pol: "It appears that this
sequence is related to the generalized heptagonal numbers A085787 in the
same way as the partition numbers A000041 are related to the generalized
pentagonal numbers A001318. ..." Therefore use the generalized heptagonal
numbers A085787 to define the H3 transform and similarly A057569 for the H1
transform. Pol's note translates to H3(A036820) = 0.
I found the following unproved results by experimentation.

With p(0) = 0,
MM(A000041) = A257628 or shift -A010815; support is A001318.
H3(A000041) = A003106-A113429
With p(0) = 1,
MM(A000041) = zero sequence,
H1(A000041) = A003114,
H3(A000041) = A003106

With sigma(0) = 0,
MM(A000203) = A147843 = -n A010815(n)
With sigma(0) = 1,
MM(A000203) = A184365
MM(A000219) = A052847
MM(A000005) = A238133
MM(A000293) = A002836

Here char means the characteristic function of a set.
MM(A000009) has support A152749 or 2 * A001318
H1(A000009) has support
15 x^2+8 x+1,
15 x^2-8 x+1,
15 x^2-2 x,
15 x^2+2 x)
H3(A000009) has support
15 x^2-16 x+4,
15 x^2-14 x+3,
15 x^2-4 x,
15 x^2+4 x)

With A000700(0) = 1,
MM(A000700) involves A001318, A058331
H1(A000700) and H3(A000700) may be similar
With A000700(0) = 0,
MM(A000700) = A089798
H1(A000700) and H3(A000700) may be similar

MM(A003114) = A113428; support A057569
H1(A003114) = A113681
H3(A003114) = 5 * A001318

MM(A003106) = A113429
H1(A003106) = 5 * (shift A001318)
H3(A003106) = A116915

H1(A036820) = A003823
H1(A036820) = zero sequence

MM(A010815) = A002107
H1 and H3 may be workable

MM(A082775) = {0, 2, 3, 4, ...}

MM(A093802) = A052905
H1(A093802) = 4 * A001318

MM(A007690) = A109389

MM(A109389) has support: A000217

These are mentioned above but I haven't tried the transforms on them:
A257628, A001318, A147843, A184365, A238133, A152749, A144459, A058331,
A089798, A113428, A113681, A057569, A003823, A002107, A052905, A000217

 MM seems to like variants of p and q; there are many such sequences in

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