A048238 Duplicate of A048140.
1, 2, 4, 8, 14, 27, 47, 86, 149, 261, 444, 760, 1269, 2119, 3486, 5711, 9247, 14906
Offset: 1
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From _M. F. Hasler_, Oct 01 2018: (Start) 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)
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);
A005987 = CoefficientList[s, x];
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;
a[n_] := (A048141[[n]] + 3 a048140[n] - a219[n] + 2 A048142[[n]])/3;
a /@ Range[0, nmax] (* Jean-François Alcover, Dec 28 2019 *)
nmax = 150;
a219[0] = 1;
a219[n_] := a219[n] = Sum[a219[n - j] DivisorSigma[2, j], {j, n}]/n;
s = Product[1/(1 - x^(2 i - 1))/(1 - x^(2 i))^Floor[i/2], {i, 1, Ceiling[( nmax + 1)/2]}] + O[x]^( nmax + 1);
A005987 = CoefficientList[s, x];
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[1] = 0;
a[n_] := (A048141[[n]] - 3 a048140[n] + 2 a219[n] - A048142[[n]])/3;
a /@ Range[1, nmax] (* Jean-François Alcover, Dec 28 2019 *)
nmax = 150;
a219[0] = 1;
a219[n_] := a219[n] = Sum[a219[n - j] DivisorSigma[2, j], {j, n}]/n;
s = Product[1/(1 - x^(2 i - 1))/(1 - x^(2 i))^Floor[i/2], {i, 1, Ceiling[( nmax + 1)/2]}] + O[x]^( nmax + 1);
A005987 = CoefficientList[s, x];
a048140[n_] := (a219[n] + A005987[[n + 1]])/2;
A048141 = Cases[Import["https://oeis.org/A048141/b048141.txt", "Table"], {, }][[All, 2]];
a[1] = 0;
a[n_] := -A048141[[n]] + 2 a048140[n] - a219[n];
a /@ Range[1, nmax] (* Jean-François Alcover, Dec 28 2019 *)
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