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A048140 Number of planar partitions of n, but partitions that are mirror images of each other (when regarded as 3-D objects) are counted only once.

%I A048140 #14 Dec 28 2019 07:46:06
%S A048140 1,2,4,8,14,27,47,86,149,261,444,760,1269,2119,3486,5711,9247,14906,
%T A048140 23800,37816,59622,93528,145759,226071,348612,535131,817280,1242824,
%U A048140 1881310,2836377,4258509,6369669,9491142,14092537,20851146,30749471
%N A048140 Number of planar partitions of n, but partitions that are mirror images of each other (when regarded as 3-D objects) are counted only once.
%C A048140 Plane partitions seen as 3-dimensional-objects can have a mirror symmetry plane.
%H A048140 Jean-François Alcover, <a href="/A048140/b048140.txt">Table of n, a(n) for n = 1..1000</a>
%e A048140 n=3 gives 4 forms: {{3}}; {{1,1,1}}={{1},{1},{1}}; {{2,1}}={{2},{1}}; {{1,1},{1}}.
%t A048140 terms = 100;
%t A048140 a219[0] = 1;
%t A048140 a219[n_] := a219[n] = Sum[a219[n - j] DivisorSigma[2, j], {j, n}]/n;
%t A048140 s = Product[1/(1 - x^(2i - 1))/(1 - x^(2i))^Floor[i/2], {i, 1, Ceiling[ (terms+1)/2]}] + O[x]^(terms+1);
%t A048140 A005987 = CoefficientList[s, x];
%t A048140 a[n_] := (a219[n] + A005987[[n+1]])/2;
%t A048140 a /@ Range[terms] (* _Jean-François Alcover_, Dec 28 2019 *)
%Y A048140 Equals (A000219+A005987)/2.
%Y A048140 Equals 2 Cs + 3 C1 + C3 + C3v, Cs=A000784, C1=A000785, C3=A048142, C3v=A048141. Cf. A000219, A005987.
%K A048140 nonn
%O A048140 1,2
%A A048140 _Wouter Meeussen_
%E A048140 Edited by _N. J. A. Sloane_ at the suggestion of _Andrew S. Plewe_, Jun 08 2007