A000784 Number of symmetrical planar partitions of n (planar partitions (A000219) that when regarded as 3-D objects have just one symmetry plane). (Formerly M0322 N0119)

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

1,3

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.

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 *)

Crossrefs

Equals -A048141 + 2*A048140 - A000219.

Cf. A000785, A000786, A005987, A048142.

Sequence in context: A055388 A065457 A280366 * A355573 A092991 A355649

Adjacent sequences: A000781 A000782 A000783 * A000785 A000786 A000787

Keyword

nonn,nice

Author

N. J. A. Sloane

Extensions

More terms from Wouter Meeussen

Status

approved