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Solid partitions are generalizations of plane partitions. MacMahon (1960) conjectured the generating function for the number of solid partitions was
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but this was subsequently shown to disagree at 
(Atkin et al. 1967). Knuth (1970) extended the tabulation
of values, but was unable to find a correct generating function. The first few values
are 1, 4, 10, 26, 59, 140, ... (OEIS A000293).
-Dimensional Partitions." Proc. Cambridge Philos. Soc. 63,
1097-1100, 1967.Knuth, D. E. "A Note on Solid Partitions."
Math. Comput. 24, 955-961, 1970.MacMahon, P. A. "Memoir
on the Theory of the Partitions of Numbers. VI: Partitions in Two-Dimensional Space,
to which is Added an Adumbration of the Theory of Partitions in Three-Dimensional
Space." Phil. Trans. Roy. Soc. London Ser. A 211, 345-373, 1912.MacMahon,
P. A. Combinatory
Analysis, Vol. 2. New York: Chelsea, pp. 75-176, 1960.Sloane,
N. J. A. Sequence A000293/M3392
in "The On-Line Encyclopedia of Integer Sequences."Weisstein, Eric W. "Solid Partition." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SolidPartition.html