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A381265 a(n) is the number of solid (3D) partitions of n with 2 layers and second layer a plane partition of 3.

%I A381265 #8 Feb 18 2025 19:00:44
%S A381265 6,21,57,138,294,606,1170,2208,4008,7176,12492,21510,36348,60801,
%T A381265 100281,164019,265263,425853
%N A381265 a(n) is the number of solid (3D) partitions of n with 2 layers and second layer a plane partition of 3.
%C A381265 Conjecture: equal to 3*(2*A000219 -A000990 -2*A000041 +1) tested up to n=20.
%H A381265 Wouter Meeussen, <a href="/A094504/a094504_1.txt">Mma functions for plane and solid partitions</a>
%F A381265 G.f.: 3*(2*Product_{k>0} 1/(1-x^k)^k -(1-x)*Product_{k>0} 1/(1-x^k)^2 - 2*Product_{k>0} 1/(1-x^k) + 1/(1 - x)) (conjectured).
%e A381265 a(3)=6 since the 6 solid partitions of {3,3} are:
%e A381265   z[{{3}},{{3}}],
%e A381265   z[{{2,1}},{{2,1}}],
%e A381265   z[{{1,1,1}},{{1,1,1}}],z[{{2},{1}},{{2},{1}}],
%e A381265   z[{{1,1},{1}},{{1,1},{1}}],
%e A381265   z[{{1},{1},{1}},{{1},{1},{1}}].
%t A381265 Table[Length@solidformBTK[{n,3}],{n,3,20}] (* or *)
%t A381265 g=20;3 CoefficientList[Series[2/Product[(1-x^m)^m,{m,g}]+ 1/(1-x)-(1-x)/Product[(1-x^m)^2,{m,g}]-2/Product[(1-x^m),{m,g}],{x,0,g}],x]
%Y A381265 Cf. A000041, A000219, A000990, A378854.
%K A381265 nonn,more
%O A381265 3,1
%A A381265 _Wouter Meeussen_, Feb 18 2025