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%I M3858 N1580 #52 Jan 25 2019 09:29:05
%S 1,5,15,45,120,326,835,2145,5345,13220,32068,76965,181975,425490,
%T 982615,2245444,5077090,11371250,25235790,55536870,121250185,
%U 262769080,565502405,1209096875,2569270050,5427963902,11404408525,23836421895,49573316740,102610460240
%N Number of 4-dimensional partitions of n.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Suresh Govindarajan, <a href="/A000334/b000334.txt">Table of n, a(n) for n = 1..40</a>
%H A. O. L. Atkin, P. Bratley, I. G. McDonald and J. K. S. McKay, <a href="/A000219/a000219.pdf">Some computations for m-dimensional partitions</a>, Proc. Camb. Phil. Soc., 63 (1967), 1097-1100. [Annotated scanned copy], <a href="http://dx.doi.org/10.1017/S0305004100042171">DOI</a>
%H S. Balakrishnan, S. Govindarajan and N. S. Prabhakar, <a href="http://arxiv.org/abs/1105.6231">On the asymptotics of higher-dimensional partitions</a>, arXiv:1105.6231 [cond-mat.stat-mech], 2011.
%H S. P. Naveen, <a href="http://www.physics.iitm.ac.in/~suresh/theses/NaveenThesis.pdf">On The Asymptotics of Some Counting Problems in Physics</a>, Thesis, Bachelor of Technology, Department of Physics, Indian Institute of Technology, Madras, May 2011.
%e From _Gus Wiseman_, Jan 23 2019: (Start)
%e The a(1) = 1 through a(3) = 15 four-dimensional partitions, represented as chains of chains of chains of integer partitions:
%e (((1))) (((2))) (((3)))
%e (((11))) (((21)))
%e (((1)(1))) (((111)))
%e (((1))((1))) (((2)(1)))
%e (((1)))(((1))) (((11)(1)))
%e (((2))((1)))
%e (((1)(1)(1)))
%e (((11))((1)))
%e (((2)))(((1)))
%e (((1)(1))((1)))
%e (((11)))(((1)))
%e (((1))((1))((1)))
%e (((1)(1)))(((1)))
%e (((1))((1)))(((1)))
%e (((1)))(((1)))(((1)))
%e (End)
%t trans[x_]:=If[x=={},{},Transpose[x]];
%t 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]}]];
%t Table[Length[levptns[n,4]],{n,8}] (* _Gus Wiseman_, Jan 24 2019 *)
%Y Cf. A000219 (2-dim), A000293 (3-dim), A000390 (5-dim), A096751 (k-dim).
%Y Cf. A002974, A007714, A050340.
%K nonn,nice
%O 1,2
%A _N. J. A. Sloane_
%E More terms from _Sean A. Irvine_, Nov 14 2010
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