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A256010 Product of n and the total number of parts in all partitions of n. Also, product of n and the sum of largest parts of all partitions of n.

%I A256010 #34 Jul 14 2015 16:43:59
%S A256010 0,1,6,18,48,100,210,378,688,1152,1920,3025,4788,7228,10920,16020,
%T A256010 23408,33405,47592,66462,92600,127092,173778,234738,316176,421275,
%U A256010 559572,736938,967260,1260137,1636890,2112185,2717664,3477078,4435708,5630660,7128504,8984044,11293638,14140893,17661840,21980264,27291222
%N A256010 Product of n and the total number of parts in all partitions of n. Also, product of n and the sum of largest parts of all partitions of n.
%C A256010 a(n) is also the volume of a three-dimensional model of partitions which is a polycube puzzle that contains n sections and A000041(n) pieces related to the A000041(n) regions of the set of partitions of n. The volume is equivalent to a(n) unit cubes.
%F A256010 a(n) = n * A006128(n).
%e A256010 For n = 6 the total number of parts in all partitions of 6 is equal to 35 so a(n) = 6 * 35 = 210. On the other hand, the sum of largest parts of all partitions of 6 is 1 + 2 + 3 + 2 + 4 + 3 + 5 + 2 + 4 + 3 + 6 = 35, so a(6) is also 6 * 35 = 210.
%e A256010 Illustration of three views of a three-dimensional model of partitions after 6th stage:
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%e A256010 For n = 6 the areas of the views are A006128(6) = 35, A066186(6) = 6 * 11 = 66 and A000290(6) = 6^2 = 36. The structure contains A000041(6) = 11 pieces and the volume is equal to a(6) = 6 * 35 = 210.
%t A256010 lim = 42; CoefficientList[Series[Sum[n x^n Product[1/(1 - x^k), {k, n}], {n, lim}], {x, 0, lim}], x] Range[0, lim] (* _Michael De Vlieger_, Jul 14 2015, after _N. J. A. Sloane_ at A006128 *)
%Y A256010 Cf. A000041, A000290, A006128, A066186, A135010, A141285, A194446, A206437.
%K A256010 nonn
%O A256010 0,3
%A A256010 _Omar E. Pol_, May 31 2015