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

0, 1, 2, 2, 4, 6, 6, 11, 16, 20, 28, 41, 51, 70, 93, 122, 158, 211, 266, 350, 450, 577, 730, 948, 1186, 1510, 1901, 2408, 2999, 3790, 4703, 5898, 7310, 9111, 11231, 13979, 17168, 21229, 26036, 32095, 39188, 48155, 58657, 71798, 87262, 106472, 129014

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice

References

• 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).

Links

• Jean-François Alcover, Table of n, a(n) for n = 1..150
• P. A. MacMahon, Combinatory analysis.

Crossrefs

Equals -A048141 + 2*A048140 - A000219.

Cf. A000785, A000786, A005987, A048142.

Programs

Mathematica

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 *)

Extensions

More terms from Wouter Meeussen