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A206440 Volume of the last section of the set of partitions of n from the shell model of partitions version "Boxes".

%I A206440 #32 Oct 20 2024 09:21:18
%S A206440 1,5,11,27,43,93,131,247,352,584,808,1306,1735,2643,3568,5160,6835,
%T A206440 9721,12672,17564,22832,30818,39743,53027,67594,88740,112752,145944,
%U A206440 183979,236059,295370,375208,467363,588007,728437,910339,1121009,1391083,1706003,2103013
%N A206440 Volume of the last section of the set of partitions of n from the shell model of partitions version "Boxes".
%C A206440 Since partial sums of this sequence give A066183 we have that A066183(n) is also the volume of the mentioned version of the shell model of partitions with n shells. Each part of size k has a volume equal to k^2 since each box is a cuboid whose sides have lengths: 1, k, k.
%H A206440 Alois P. Heinz, <a href="/A206440/b206440.txt">Table of n, a(n) for n = 1..10000</a>
%F A206440 a(n) ~ sqrt(3) * zeta(3) * exp(Pi*sqrt(2*n/3)) / Pi^2. - _Vaclav Kotesovec_, Oct 20 2024
%p A206440 b:= proc(n, i) option remember; `if`(n=0 or i=1, [1, n],
%p A206440       b(n, i-1)+(p-> p+[0, p[1]*i^2])(b(n-i, min(n-i, i))))
%p A206440     end:
%p A206440 a:= n-> (b(n$2)-b(n-1$2))[2]:
%p A206440 seq(a(n), n=1..40);  # _Alois P. Heinz_, Feb 23 2022
%t A206440 b[n_, i_] := b[n, i] = If[n == 0 || i == 1, {1, n},
%t A206440      b[n, i-1] + Function[p, p + {0, p[[1]]*i^2}][b[n-i, Min[n-i, i]]]];
%t A206440 a[n_] := (b[n, n] - b[n-1, n-1])[[2]];
%t A206440 Table[a[n], {n, 1, 40}] (* _Jean-François Alcover_, Apr 25 2022, after _Alois P. Heinz_ *)
%Y A206440 Row sums of triangle A206438. Partial sums give A066183.
%Y A206440 Cf. A135010, A138121, A141285.
%K A206440 nonn
%O A206440 1,2
%A A206440 _Omar E. Pol_, Feb 08 2012