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A328041 Number of parts in all proper floor(n/2)-times partitions of n.

%I A328041 #10 Dec 18 2020 04:02:29
%S A328041 0,1,2,5,21,61,461,1652,17487,76264,1002835,5207742,88664398,
%T A328041 515821495,10184805624,69200406679,1610282904928,12024183111167,
%U A328041 318978837371853,2653055962437988,79332250069994262,725413309833320933,23919660963588169669,238830233430136549070
%N A328041 Number of parts in all proper floor(n/2)-times partitions of n.
%H A328041 Alois P. Heinz, <a href="/A328041/b328041.txt">Table of n, a(n) for n = 0..300</a>
%F A328041 a(n) = A327631(n,floor(n/2)).
%p A328041 b:= proc(n, i, k) option remember; `if`(n=0, [1, 0],
%p A328041      `if`(k=0, [1, 1], `if`(i<2, 0, b(n, i-1, k))+
%p A328041          (h-> (f-> f +[0, f[1]*h[2]/h[1]])(h[1]*
%p A328041         b(n-i, min(n-i, i), k)))(b(i$2, k-1))))
%p A328041     end:
%p A328041 a:= n-> (k-> add(b(n$2, i)[2]*(-1)^(k-i)
%p A328041         *binomial(k, i), i=0..k))(iquo(n,2)):
%p A328041 seq(a(n), n=0..23);
%t A328041 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 A328041 a[n_] := With[{k = Quotient[n, 2]}, Sum[b[n, n, i][[2]] (-1)^(k - i)* Binomial[k, i], {i, 0, k}]];
%t A328041 a /@ Range[0, 23] (* _Jean-François Alcover_, Dec 18 2020, after _Alois P. Heinz_ *)
%Y A328041 Cf. A327631.
%K A328041 nonn
%O A328041 0,3
%A A328041 _Alois P. Heinz_, Oct 02 2019