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A000293
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a(n) = number of solid (i.e., three-dimensional) partitions of n.
(Formerly M3392 N1371)
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37
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1, 1, 4, 10, 26, 59, 140, 307, 684, 1464, 3122, 6500, 13426, 27248, 54804, 108802, 214071, 416849, 805124, 1541637, 2930329, 5528733, 10362312, 19295226, 35713454, 65715094, 120256653, 218893580, 396418699, 714399381, 1281403841, 2287986987, 4067428375, 7200210523, 12693890803, 22290727268, 38993410516, 67959010130, 118016656268, 204233654229, 352245710866, 605538866862, 1037668522922, 1772700955975, 3019333854177, 5127694484375, 8683676638832, 14665233966068, 24700752691832, 41495176877972, 69531305679518
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OFFSET
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0,3
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COMMENTS
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An ordinary partition is a row of numbers in nondecreasing order whose sum is n. Here the numbers are in a three-dimensional pile, nondecreasing in the x-, y- and z-directions.
Finding a g.f. for this sequence is an unsolved problem. At first it was thought that it was given by A000294.
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REFERENCES
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P. A. MacMahon, Memoir on the theory of partitions of numbers - Part VI, Phil. Trans. Roal Soc., 211 (1912), 345-373.
P. A. MacMahon, Combinatory Analysis. Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p 332.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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EXAMPLE
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Examples for n=2 and n=3.
a(2) = 4: 2; 11 where the first 1 is at the origin and the second 1 is in the x, y or z direction.
a(3) = 10: 3; 21 where the 2 is at the origin and the 1 is on the x, y or z axis; 111 (a row of 3 ones on the x, y or z axes); and three 1's with one 1 at the origin and the other two 1's on two of the three axes.
The a(1) = 1 through a(4) = 26 solid partitions, represented as chains of chains of integer partitions:
((1)) ((2)) ((3)) ((4))
((11)) ((21)) ((22))
((1)(1)) ((111)) ((31))
((1))((1)) ((2)(1)) ((211))
((11)(1)) ((1111))
((2))((1)) ((2)(2))
((1)(1)(1)) ((3)(1))
((11))((1)) ((21)(1))
((1)(1))((1)) ((11)(11))
((1))((1))((1)) ((111)(1))
((2))((2))
((3))((1))
((2)(1)(1))
((21))((1))
((11))((11))
((11)(1)(1))
((111))((1))
((2)(1))((1))
((1)(1)(1)(1))
((11)(1))((1))
((2))((1))((1))
((1)(1))((1)(1))
((1)(1)(1))((1))
((11))((1))((1))
((1)(1))((1))((1))
((1))((1))((1))((1))
(End)
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MATHEMATICA
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planePtns[n_]:=Join@@Table[Select[Tuples[IntegerPartitions/@ptn], And@@(GreaterEqual@@@Transpose[PadRight[#]])&], {ptn, IntegerPartitions[n]}];
solidPtns[n_]:=Join@@Table[Select[Tuples[planePtns/@y], And@@(GreaterEqual@@@Transpose[Join@@@(PadRight[#, {n, n}]&/@#)])&], {y, IntegerPartitions[n]}];
Table[Length[solidPtns[n]], {n, 10}] (* Gus Wiseman, Jan 23 2019 *)
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CROSSREFS
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Cf. A000041, A000219 (2-dim), A000294, A000334 (4-dim), A000390 (5-dim), A002835, A002836, A005980, A037452 (inverse Euler trans.), A080207, A007326, A000416 (6-dim), A000427 (7-dim), A179855 (8-dim).
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KEYWORD
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nonn,nice
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AUTHOR
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EXTENSIONS
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More terms from the Mustonen and Rajesh article, May 02 2003
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STATUS
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approved
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