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<DIV><FONT face=Arial size=2>This is a hurried attempt to provide background on
the genesis of A123937. I am sure that a most of what I write here is a
poor rehash of basic theory, and this whole post probably equates to a few lines
of generating function theory.</FONT></DIV>
<DIV><FONT face=Arial size=2></FONT> </DIV>
<DIV><FONT face=Arial size=2>For d and n in Z+, let A_d(n) be the set of integer
points on the d-sphere of radius sqrt(n) centered at the origin, that
is:</FONT></DIV>
<DIV> </DIV>
<DIV><FONT face=Arial size=2> [1] a_d(n) = #({P in Z^d : |P|^2
= n}).</FONT></DIV>
<DIV> </DIV>
<DIV><FONT face=Arial size=2>A table of a_d(n) for a few small values of d and
n:</FONT></DIV>
<DIV> </DIV>
<DIV><FONT face=Arial size=2>Table of a_d(n)</FONT></DIV>
<DIV> </DIV>
<DIV><FONT face=Arial size=2> \d| 0
1 2 3 4
5 6
7 8
9
10<BR> n\|<BR>---+--------------------------------------------------------------<BR> 0
| 1 1 1
1 1 1
1 1
1 1
1<BR> 1 | 0 2 4
6 8 10 12
14 16
18 20<BR> 2 | 0
0 4 12 24
40 60 84
112 144 180<BR> 3 |
0 0 0 8
32 80 160 280
448 672 960<BR> 4 |
0 2 4 6
24 90 252 574 1136
2034 3380<BR> 5 | 0
0 8 24 48 112
312 840 2016 4320
8424<BR> 6 | 0 0 0
24 96 240 544 1288
3136 7392 16320<BR> 7 | 0
0 0 0 64 320
960 2368 5504 12672 28800<BR> 8 |
0 0 4 12 24
200 1020 3444 9328 22608
52020<BR> 9 | 0 2 4
30 104 250 876 3542 12112
34802 88660<BR>10 | 0 0
8 24 144 560 1560 4424 14112
44640 129064</FONT></DIV>
<DIV> </DIV>
<DIV><FONT face=Arial size=2>Each sequence a_d is a column of this table. All of
these column sequences already exist in the OEIS.</FONT></DIV>
<DIV><FONT face=Arial size=2></FONT> </DIV>
<DIV><FONT face=Arial size=2>It turns out that a_d can be gotten by convolving
the sequence j = (1,2,0,0,2,...) = A000122 d times with the convolutional
identity sequence e = (1,0,0,0,0,...) = A000007. We could say that a_d = j^d as
a "convolutional power".</FONT></DIV>
<DIV> </DIV>
<DIV><FONT face=Arial size=2>The generating function of a_d is </FONT><FONT
face=Arial size=2>t(x)^d where t is the generating function of j. t turns out to
be the Jacobi theta function theta_3, as noted on A000122. </FONT><FONT
face=Arial size=2>If we apply the finite difference transform to the sequence of
generating functions </FONT><FONT face=Arial size=2>{t</FONT><FONT face=Arial
size=2>(x)^d}, we obtain the sequence</FONT><FONT face=Arial
size=2> {(t(x)-1)^d}. Expanding these functions back to sequences gives the
sequence of sequences {b_d} where b_d = (j-e)^d as a convolutional power. This
allows us to constuct a table of b_d:</FONT></DIV>
<DIV> </DIV>
<DIV><FONT face=Arial size=2>Table of b_d(n)</FONT></DIV>
<DIV> </DIV>
<DIV><FONT face=Arial size=2> \d| 0
1 2 3 4
5 6 7 8
9
10<BR> n\|<BR>---+------------------------------------------------------<BR> 0
| 1 0 0
0 0 0 0
0 0 0 0<BR> 1
| 0 2 0
0 0 0 0
0 0 0 0<BR> 2
| 0 0 4
0 0 0 0
0 0 0 0<BR> 3
| 0 0 0
8 0 0 0
0 0 0 0<BR> 4
| 0 2 0
0 16 0 0
0 0 0 0<BR> 5
| 0 0 8
0 0 32 0
0 0 0 0<BR> 6
| 0 0 0
24 0 0 64
0 0 0 0<BR> 7
| 0 0 0
0 64 0 0
128 0 0 0<BR> 8
| 0 0 4
0 0 160 0 0
256 0 0<BR> 9 |
0 2 0 24
0 0 384 0 0
512 0<BR>10 | 0
0 8 0 96
0 0 896 0 0
1024</FONT></DIV>
<DIV> </DIV>
<DIV><FONT face=Arial size=2>It turns out that the process we used to get from
the a_d table to the b_d table, namely, converting columns of a_d to generating
functions, applying the finite difference transform to the gf's, and expanding
the resulting functions</FONT><FONT face=Arial size=2> back to columns of
b_d, is the same as applying the finite difference transform to the rows of the
a_d table to obtain rows of the b_d table.</FONT></DIV>
<DIV><FONT face=Arial size=2></FONT> </DIV>
<DIV><FONT face=Arial size=2>An inductive proof starting with b_d = (j-e)^d
where </FONT><FONT face=Arial size=2>j-e = (0,2,0,0,2,...) shows that b_d(d) =
2^d and b_d(n) = 0 for n < d. This means that the nth element of the nth row
of the b_d table is the last nonzero element in that row, which in turn implies
that the nth row of the a_d table, that is, the number of integer points P with
|P|^2 = n, is a polynomial of degree n in d.</FONT></DIV>
<DIV><FONT face=Arial size=2></FONT> </DIV>
<DIV><FONT face=Arial size=2>If, instead of [1], we had </FONT><FONT face=Arial
size=2>started with</FONT></DIV>
<DIV> </DIV>
<DIV><FONT face=Arial size=2> [1] a_d'(n) = #({P in Z^d :
|P|^2 <= n})</FONT></DIV>
<DIV> </DIV>
<DIV><FONT face=Arial size=2>which counts points inside as well as on the
d-sphere, the nth row of a_d' table </FONT><FONT face=Arial size=2>would be
gotten by adding termwise the rows 0 through n of the a_d table. The sequence
a_d' has generating function t(x)^d/(1-x). Similarly, the nth row of the
corresponding b_d' table </FONT><FONT face=Arial size=2>is the sum of rows 0
through n of the b_d table, and sequence b_d has generating function
(t(x)-1)^d)/(1-x). The table of b_d', omitting the superdiagonal zeroes, is
sequence A123937.<BR></DIV></FONT>
<DIV><FONT face=Arial size=2>The recurrence given in the title of A123937
follows from the relation</FONT></DIV><FONT face=Arial size=2>
<DIV><BR> b_d = (1,1,1,1,1,...) = A000012 if d = 0; (j-e) *
b_{d-1}' if d >= 1.</DIV>
<DIV> </DIV>
<DIV>I know this is a spotty exposition, but I'm sure the loose ends could be
tied together by a real number theorist.</FONT></DIV>
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