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A000786
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Number of inequivalent planar partitions of n, when considering them as 3D objects.
(Formerly M1020 N0383)
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11
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1, 1, 1, 2, 4, 6, 11, 19, 33, 55, 95, 158, 267, 442, 731, 1193, 1947, 3137, 5039, 8026, 12726, 20024, 31373, 48835, 75673, 116606, 178889, 273061, 415086, 628115, 946723, 1421082, 2125207, 3166152, 4700564, 6954151, 10254486, 15071903
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OFFSET
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0,4
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COMMENTS
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Partitions that are the same when regarded as 3-D objects are counted only once. - Wouter Meeussen, May 2006
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REFERENCES
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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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FORMULA
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EXAMPLE
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For n = 2, all three plane partitions [2], [1 1] and [1; 1] (where ";" means next row) correspond to a 1 X 1 X 2 rectangular cuboid, therefore a(2) = 1.
For n = 3, we have [3] ~ [1 1 1] ~ [1; 1; 1] all corresponding to a 1 X 1 X 3 rectangular cuboid or tower of height 3, and [2 1] ~ [2; 1] ~ [1 1; 1] correspond to an L-shaped object, therefore a(3) = 2.
For n = 4, [4] ~ [1 1 1 1] ~ [1; 1; 1; 1] correspond to the 4-tower; [3 1] ~ [3; 1] ~ [2 1 1] ~ [2; 1; 1] ~ [1 1 1; 1] ~ [1 1; 1; 1] all correspond to the same L-shaped object, [2 2] ~ [2; 2] ~ [1 1; 1 1] represent a "flat" square, and it remains [2, 1; 1], so a(4) = 4.
For n = 5, we again have the tower [5] ~ [1 1 1 1 1] ~ [1; 1; 1; 1; 1], a "narrow L" or 4-tower with one "foot" [4 1] ~ [4; 1] ~ [2 1 1 1] ~ [2; 1; 1; 1] ~ [1 1 1 1; 1] ~ [1 1; 1; 1; 1], a symmetric L-shape [3 1 1] ~ [3; 1; 1] ~ [1 1 1; 1; 1], a 3-tower with 2 feet [3 1; 1] ~ [2 1; 1; 1] ~ [2 1 1; 1], a flat 2+3 shape [3 2] ~ [3; 2] ~ [2 2 1] ~ [2; 2; 1] ~ [1 1 1; 1 1] ~ [1 1; 1 1; 1] and a 2X2 square with a cube on top, [2 1;1 1] ~ [2 2; 1] ~ [2 1; 2]. This yields a(5) = 6 classes. (End)
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MATHEMATICA
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nmax = 150;
a219[0] = 1;
a219[n_] := a219[n] = Sum[a219[n - j] DivisorSigma[2, j], {j, n}]/n;
s = Product[1/(1 - x^(2i - 1))/(1 - x^(2i))^Floor[i/2], {i, 1, Ceiling[( nmax+1)/2]}] + O[x]^(nmax+1);
a048140[n_] := (a219[n] + A005987[[n+1]])/2;
A048141 = Cases[Import["https://oeis.org/A048141/b048141.txt", "Table"], {_, _}][[All, 2]];
A048142 = Cases[Import["https://oeis.org/A048142/b048142.txt", "Table"], {_, _}][[All, 2]];
a[0] = 1;
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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EXTENSIONS
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Name & links edited and a(0) = 1 added by M. F. Hasler, Sep 30 2018
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STATUS
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approved
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