A000785 Number of asymmetrical planar partitions of n: planar partitions (A000219) that when regarded as 3-D objects have no symmetry. (Formerly M1392 N0542)

0, 0, 0, 1, 2, 5, 11, 21, 39, 73, 129, 226, 388, 659, 1100, 1821, 2976, 4828, 7754, 12370, 19574, 30789, 48097, 74725, 115410, 177366, 271159, 412665, 625098, 942932, 1416362, 2119282, 3158840, 4691431, 6942882, 10240503, 15054705

Offset

1,5

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]];
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 *)

Crossrefs

Equals (A048141 - 3*A048140 + 2*A000219 - A048142)/3.

Cf. A000784, A000786, A005987.

Sequence in context: A082775 A023548 A144700 * A364552 A369525 A364591

Adjacent sequences: A000782 A000783 A000784 * A000786 A000787 A000788

Keyword

nonn,nice

Author

N. J. A. Sloane

Extensions

More terms from Wouter Meeussen

Status

approved