a possible link to solid partitions

[wouter meeussen]

A000293 Sequence :
 1, 1, 4, 10, 26, 59, 140, 307, 684, 1464, 3122, 6500, 13426, 27248,
54804, 108802, 214071 ...
Comment: Finding a GF for this sequence is a famous unsolved problem.

the table A089353 :
Triangle read by rows: T(n,m) = number of planar partitions of n with
trace m.

1,
2,1,
3,2,1,
4,6,2,1,
5,10,6,2,1,
6,19,14,6,2,1
...

can be coded to a set of polynomials in y,
each of which can be set equal to A000293[n]
and solved for the y[1],..,y[n] as below:

Solve[{
y[1] == 1,
2 y[1] + y[2] == 4,
3 y[1] + 2 y[2] + y[3] == 10,
4 y[1] + 6 y[2] + 2 y[3] + y[4] == 26,
5 y[1] + 10 y[2] + 6 y[3] + 2 y[4] + y[5] == 59
}, {y[1], y[2], y[3], y[4], y[5]}]

This amounts to building up the solid partitions from the plane partitions.

I get for y[1],..,y[n] :
1, 2, 3, 4, 8, 14, 28, 48, 76, 124, 215, 352, 659, 1142, 2137, 3697, 6642,
11200, 19389, 32330, 54870, 91429, 154342, 257121, 432584, 720545, 1206326,
2003522, 3334326, 5508846, 9110472, 14971258

This kind of decent, monotonicly rising sequence screams "GF",
so I fed it to SuperSeeker. Alas, to no avail.

But humans of the mathematical 'penchant' can do more than SuSeek;
Any takers?

W.