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A000784 Number of symmetrical planar partitions of n (planar partitions (A000219) that when regarded as 3-D objects have just one symmetry plane).

%I A000784 M0322 N0119 #24 Jun 17 2022 14:51:26
%S A000784 0,1,2,2,4,6,6,11,16,20,28,41,51,70,93,122,158,211,266,350,450,577,
%T A000784 730,948,1186,1510,1901,2408,2999,3790,4703,5898,7310,9111,11231,
%U A000784 13979,17168,21229,26036,32095,39188,48155,58657,71798,87262,106472,129014
%N A000784 Number of symmetrical planar partitions of n (planar partitions (A000219) that when regarded as 3-D objects have just one symmetry plane).
%D A000784 P. A. MacMahon, Combinatory Analysis. Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p 332.
%D A000784 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000784 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000784 Jean-François Alcover, <a href="/A000784/b000784.txt">Table of n, a(n) for n = 1..150</a>
%H A000784 P. A. MacMahon, <a href="http://www.hti.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=ABU9009">Combinatory analysis</a>.
%t A000784 nmax = 150;
%t A000784 a219[0] = 1;
%t A000784 a219[n_] := a219[n] = Sum[a219[n - j] DivisorSigma[2, j], {j, n}]/n;
%t A000784 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);
%t A000784 A005987 = CoefficientList[s, x];
%t A000784 a048140[n_] := (a219[n] + A005987[[n + 1]])/2;
%t A000784 A048141 = Cases[Import["https://oeis.org/A048141/b048141.txt", "Table"], {_, _}][[All, 2]];
%t A000784 a[1] = 0;
%t A000784 a[n_] := -A048141[[n]] + 2 a048140[n] - a219[n];
%t A000784 a /@ Range[1, nmax] (* _Jean-François Alcover_, Dec 28 2019 *)
%Y A000784 Equals -A048141 + 2*A048140 - A000219.
%Y A000784 Cf. A000785, A000786, A005987, A048142.
%K A000784 nonn,nice
%O A000784 1,3
%A A000784 _N. J. A. Sloane_
%E A000784 More terms from _Wouter Meeussen_