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A328042 Number of parts in all proper twice partitions of n.

%I A328042 #13 Dec 10 2020 03:27:42
%S A328042 3,21,61,205,474,1246,2723,6277,12961,28682,56976,118919,234715,
%T A328042 473988,913011,1807211,3430048,6648397,12500170,23764885,44174088,
%U A328042 83090853,152803509,283387971,517516615,949775754,1719088271,3127641937,5618833687,10133255636
%N A328042 Number of parts in all proper twice partitions of n.
%H A328042 Vaclav Kotesovec, <a href="/A328042/b328042.txt">Table of n, a(n) for n = 3..5000</a> (terms 3..2000 from Alois P. Heinz)
%p A328042 b:= proc(n, i, k) option remember; `if`(n=0, [1, 0],
%p A328042      `if`(k=0, [1, 1], `if`(i<2, 0, b(n, i-1, k))+
%p A328042          (h-> (f-> f +[0, f[1]*h[2]/h[1]])(h[1]*
%p A328042         b(n-i, min(n-i, i), k)))(b(i$2, k-1))))
%p A328042     end:
%p A328042 a:= n-> (k-> add(b(n$2, i)[2]*(-1)^(k-i)*binomial(k, i), i=0..k))(2):
%p A328042 seq(a(n), n=3..35);
%t A328042 b[n_, i_, k_] := b[n, i, k] = If[n == 0, {1, 0}, If[k == 0, {1, 1}, If[i < 2, 0, b[n, i - 1, k]] + Function[h, Function[f, f + {0, f[[1]] h[[2]]/ h[[1]]}][h[[1]] b[n - i, Min[n - i, i], k]]][b[i, i, k - 1]]]];
%t A328042 a[n_] := With[{k = 2}, Sum[b[n, n, i][[2]] (-1)^(k-i) Binomial[k, i], {i, 0, k}]];
%t A328042 a /@ Range[3, 35] (* _Jean-François Alcover_, Dec 10 2020, after _Alois P. Heinz_ *)
%Y A328042 Column k=2 of A327631.
%Y A328042 Cf. A327769.
%K A328042 nonn
%O A328042 3,1
%A A328042 _Alois P. Heinz_, Oct 02 2019