[seqfan] A019298 and interesting identity.

[L. Edson Jeffery]

Hello,

For m >= 1, denote by S_m(n) the number of m X m symmetric matrices of
nonnegative integers in which every row (and column) sums to n and, for k
>= 3, let Y_k(n) denote the number of magic labelings (in nonnegative
integers) of the prism graph Y_k (or I X C_k, depending on who defines it)
with magic sum n.

Using Macmahan's omega operators in the "Omega" package for Mathematica
(authored by Axel Riese), it was not hard to prove that

(1)  S_3(n) = Y_3(n), n = 0,1,2,...,

by showing that the sequences {S_3(n)} and {Y_3(n)} have the same
generating function (see https://oeis.org/A019298). The graph Y_3 is the
one on the top left in the image: http://mathworld.wolfram.com/
images/eps-gif/PrismGraph_1000.gif.

Is there another argument, possibly geometric, that explains why (1) is
true?

One of the reasons why this interests me is because the sequences {S_k(n)}
and {Y_k(n)} appear to not coincide for any k > 3. Another reason is
because of the following.

Let M_n, n >= 1, be the n X n matrix

M_1 = {{1}}  (n=1),

M_n = {{0,...,0,1}, {0,...,0,1,1}, ..., {0,1,...,1}, {1,...,1}}  (n>1),

and consider the following array A starting as:

{ 1, 1,  1,   1,    1,     1,     1,      1,       1,        1,         1,
...},
{ 2, 1,  3,   4,    7,    11,    18,     29,      47,       76,       123,
...},
{ 3, 2,  6,  11,   26,    57,   129,    289,     650,     1460,      3281,
...},
{ 4, 2, 10,  23,   70,   197,   571,   1640,    4726,    13604,     39175,
...},
{ 5, 3, 15,  42,  155,   533,  1884,   6604,   23219,    81555,    286555,
...},
{ 6, 3, 21,  69,  301,  1223,  5103,  21122,   87677,   363606,   1508401,
...},
{ 7, 4, 28, 106,  532,  2494, 11998,  57271,  274132,  1310974,   6271378,
...},
{ 8, 4, 36, 154,  876,  4654, 25362, 137155,  743724,  4029310,  21836366,
...},
{ 9, 5, 45, 215, 1365,  8105, 49347, 298184, 1806597, 10936124,  66220705,
...},
{10, 5, 55, 290, 2035, 13355, 89848, 599954, 4016683, 26868719, 179784715,
...},
....

The entry A(n,k) in row n >= 1 and oolumn k >= 0 of A is given by

(2)  A(n,k) = tr((A_n)^k) = Sum_{j=1..n} s(n-1,x(n,j))^k,

where tr(..) denotes the trace,

x(n,j) = 2*(-1)^(j+1)*cos(j*Pi/(2*n+1)),

s(0,x) = 1, s(1,x) = x and s(n,x) = x*s(n-1,x) - s(n-2,x) (n>1)

(these are Wolfdieter Lang's S-polynomials).

It appears, for k >= 3, that column k of A is the sequence {Y_k(n-1)}. (At
least for columns k = 3..8, respectively, the terms match those in A019298,
A006325, A244497, A244879, A244873, A244880, respectively.)

Can anyone prove or disprove this conjecture?

Ed Jeffery