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A000334
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Number of 4-dimensional partitions of n.
(Formerly M3858 N1580)
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11
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1, 5, 15, 45, 120, 326, 835, 2145, 5345, 13220, 32068, 76965, 181975, 425490, 982615, 2245444, 5077090, 11371250, 25235790, 55536870, 121250185, 262769080, 565502405, 1209096875, 2569270050, 5427963902, 11404408525, 23836421895, 49573316740, 102610460240
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OFFSET
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1,2
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REFERENCES
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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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Suresh Govindarajan, Table of n, a(n) for n = 1..40
A. O. L. Atkin, P. Bratley, I. G. McDonald and J. K. S. McKay, Some computations for m-dimensional partitions, Proc. Camb. Phil. Soc., 63 (1967), 1097-1100. [Annotated scanned copy], DOI
S. Balakrishnan, S. Govindarajan and N. S. Prabhakar, On the asymptotics of higher-dimensional partitions, arXiv:1105.6231 [cond-mat.stat-mech], 2011.
S. P. Naveen, On The Asymptotics of Some Counting Problems in Physics, Thesis, Bachelor of Technology, Department of Physics, Indian Institute of Technology, Madras, May 2011.
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EXAMPLE
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From Gus Wiseman, Jan 23 2019: (Start)
The a(1) = 1 through a(3) = 15 four-dimensional partitions, represented as chains of chains of chains of integer partitions:
(((1))) (((2))) (((3)))
(((11))) (((21)))
(((1)(1))) (((111)))
(((1))((1))) (((2)(1)))
(((1)))(((1))) (((11)(1)))
(((2))((1)))
(((1)(1)(1)))
(((11))((1)))
(((2)))(((1)))
(((1)(1))((1)))
(((11)))(((1)))
(((1))((1))((1)))
(((1)(1)))(((1)))
(((1))((1)))(((1)))
(((1)))(((1)))(((1)))
(End)
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MATHEMATICA
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trans[x_]:=If[x=={}, {}, Transpose[x]];
levptns[n_, k_]:=If[k==1, IntegerPartitions[n], Join@@Table[Select[Tuples[levptns[#, k-1]&/@y], And@@(GreaterEqual@@@trans[Flatten/@(PadRight[#, ConstantArray[n, k-1]]&/@#)])&], {y, IntegerPartitions[n]}]];
Table[Length[levptns[n, 4]], {n, 8}] (* Gus Wiseman, Jan 24 2019 *)
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CROSSREFS
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Cf. A000219 (2-dim), A000293 (3-dim), A000390 (5-dim), A096751 (k-dim).
Cf. A002974, A007714, A050340.
Sequence in context: A094283 A158875 A022813 * A000335 A271180 A200465
Adjacent sequences: A000331 A000332 A000333 * A000335 A000336 A000337
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KEYWORD
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nonn,nice
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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More terms from Sean A. Irvine, Nov 14 2010
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STATUS
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approved
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