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A339420 Number of compositions (ordered partitions) of n into an even number of cubes.

%I A339420 #10 Dec 03 2020 18:34:28
%S A339420 1,0,1,0,1,0,1,0,1,2,1,4,1,6,1,8,2,10,7,12,16,14,29,16,46,22,67,40,94,
%T A339420 78,125,144,161,246,214,394,312,602,499,878,835,1236,1396,1722,2286,
%U A339420 2446,3637,3614,5598,5560,8358,8782,12226,14014,17776,22278,26056,34924
%N A339420 Number of compositions (ordered partitions) of n into an even number of cubes.
%H A339420 Alois P. Heinz, <a href="/A339420/b339420.txt">Table of n, a(n) for n = 0..5000</a>
%H A339420 <a href="/index/Com#comp">Index entries for sequences related to compositions</a>
%H A339420 <a href="/index/Su#ssq">Index entries for sequences related to sums of cubes</a>
%F A339420 G.f.: (1/2) * (1 / (1 - Sum_{k>=1} x^(k^3)) + 1 / Sum_{k>=0} x^(k^3)).
%F A339420 a(n) = (A023358(n) + A323633(n)) / 2.
%F A339420 a(n) = Sum_{k=0..n} A023358(k) * A323633(n-k).
%e A339420 a(11) = 4 because we have [8, 1, 1, 1], [1, 8, 1, 1], [1, 1, 8, 1] and [1, 1, 1, 8].
%p A339420 b:= proc(n, t) option remember; local r, f, g;
%p A339420       if n=0 then t else r, f, g:=$0..2; while f<=n
%p A339420       do r, f, g:= r+b(n-f, 1-t), f+3*g*(g-1)+1, g+1 od; r fi
%p A339420     end:
%p A339420 a:= n-> b(n, 1):
%p A339420 seq(a(n), n=0..60);  # _Alois P. Heinz_, Dec 03 2020
%t A339420 nmax = 57; CoefficientList[Series[(1/2) (1/(1 - Sum[x^(k^3), {k, 1, Floor[nmax^(1/3)] + 1}]) + 1/Sum[x^(k^3), {k, 0, Floor[nmax^(1/3)] + 1}]), {x, 0, nmax}], x]
%Y A339420 Cf. A000578, A023358, A034008, A323633, A339368, A339418, A339421.
%K A339420 nonn
%O A339420 0,10
%A A339420 _Ilya Gutkovskiy_, Dec 03 2020