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Number of asymmetrical planar partitions of n: planar partitions (A000219) that when regarded as 3-D objects have no symmetry.
(Formerly M1392 N0542)
10

%I M1392 N0542 #24 Feb 12 2023 10:02:54

%S 0,0,0,1,2,5,11,21,39,73,129,226,388,659,1100,1821,2976,4828,7754,

%T 12370,19574,30789,48097,74725,115410,177366,271159,412665,625098,

%U 942932,1416362,2119282,3158840,4691431,6942882,10240503,15054705

%N Number of asymmetrical planar partitions of n: planar partitions (A000219) that when regarded as 3-D objects have no symmetry.

%D 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 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Jean-François Alcover, <a href="/A000785/b000785.txt">Table of n, a(n) for n = 1..150</a>

%H P. A. MacMahon, <a href="http://www.hti.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=ABU9009">Combinatory analysis</a>.

%t nmax = 150;

%t a219[0] = 1;

%t a219[n_] := a219[n] = Sum[a219[n - j] DivisorSigma[2, j], {j, n}]/n;

%t 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 A005987 = CoefficientList[s, x];

%t a048140[n_] := (a219[n] + A005987[[n + 1]])/2;

%t A048141 = Cases[Import["https://oeis.org/A048141/b048141.txt", "Table"], {_, _}][[All, 2]];

%t A048142 = Cases[Import["https://oeis.org/A048142/b048142.txt", "Table"], {_, _}][[All, 2]];

%t a[1] = 0;

%t a[n_] := (A048141[[n]] - 3 a048140[n] + 2 a219[n] - A048142[[n]])/3;

%t a /@ Range[1, nmax] (* _Jean-François Alcover_, Dec 28 2019 *)

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

%Y Cf. A000784, A000786, A005987.

%K nonn,nice

%O 1,5

%A _N. J. A. Sloane_

%E More terms from _Wouter Meeussen_