cp's OEIS Frontend

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

%I A000785 M1392 N0542 #24 Feb 12 2023 10:02:54
%S A000785 0,0,0,1,2,5,11,21,39,73,129,226,388,659,1100,1821,2976,4828,7754,
%T A000785 12370,19574,30789,48097,74725,115410,177366,271159,412665,625098,
%U A000785 942932,1416362,2119282,3158840,4691431,6942882,10240503,15054705
%N A000785 Number of asymmetrical planar partitions of n: planar partitions (A000219) that when regarded as 3-D objects have no symmetry.
%D A000785 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 A000785 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000785 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000785 Jean-François Alcover, <a href="/A000785/b000785.txt">Table of n, a(n) for n = 1..150</a>
%H A000785 P. A. MacMahon, <a href="http://www.hti.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=ABU9009">Combinatory analysis</a>.
%t A000785 nmax = 150;
%t A000785 a219[0] = 1;
%t A000785 a219[n_] := a219[n] = Sum[a219[n - j] DivisorSigma[2, j], {j, n}]/n;
%t A000785 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 A000785 A005987 = CoefficientList[s, x];
%t A000785 a048140[n_] := (a219[n] + A005987[[n + 1]])/2;
%t A000785 A048141 = Cases[Import["https://oeis.org/A048141/b048141.txt", "Table"], {_, _}][[All, 2]];
%t A000785 A048142 = Cases[Import["https://oeis.org/A048142/b048142.txt", "Table"], {_, _}][[All, 2]];
%t A000785 a[1] = 0;
%t A000785 a[n_] := (A048141[[n]] - 3 a048140[n] + 2 a219[n] - A048142[[n]])/3;
%t A000785 a /@ Range[1, nmax] (* _Jean-François Alcover_, Dec 28 2019 *)
%Y A000785 Equals (A048141 - 3*A048140 + 2*A000219 - A048142)/3.
%Y A000785 Cf. A000784, A000786, A005987.
%K A000785 nonn,nice
%O A000785 1,5
%A A000785 _N. J. A. Sloane_
%E A000785 More terms from _Wouter Meeussen_