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A327647 Number of parts in all proper many times partitions of n into distinct parts.

%I A327647 #15 May 03 2020 13:42:35
%S A327647 0,1,1,3,6,15,38,133,446,1913,7492,36293,175904,953729,5053294,
%T A327647 31353825,188697696,1268175779,8356974190,61775786301,448436391810,
%U A327647 3579695446911,27848806031468,239229189529685,2019531300063238,18477179022470655,165744369451885256
%N A327647 Number of parts in all proper many times partitions of n into distinct parts.
%C A327647 In each step at least one part is replaced by the partition of itself into smaller distinct parts. The parts are not resorted and the parts in the result are not necessarily distinct.
%H A327647 Alois P. Heinz, <a href="/A327647/b327647.txt">Table of n, a(n) for n = 0..300</a>
%H A327647 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>
%e A327647 a(4) = 6 = 1 + 2 + 3, counting the (final) parts in 4, 4->31, 4->31->211.
%p A327647 b:= proc(n, i, k) option remember; `if`(n=0, [1, 0],
%p A327647      `if`(k=0, [1, 1], `if`(i*(i+1)/2<n, 0, b(n, i-1, k)+
%p A327647          (h-> (f-> f +[0, f[1]*h[2]/h[1]])(h[1]*
%p A327647         b(n-i, min(n-i, i-1), k)))(b(i$2, k-1)))))
%p A327647     end:
%p A327647 a:= n-> add(add(b(n$2, i)[2]*(-1)^(k-i)*
%p A327647         binomial(k, i), i=0..k), k=0..max(0, n-2)):
%p A327647 seq(a(n), n=0..27);
%t A327647 b[n_, i_, k_] := b[n, i, k] = If[n == 0, {1, 0}, If[k == 0, {1, 1}, If[i(i + 1)/2 < n, 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 - 1], k]]][b[i, i, k - 1]]]]];
%t A327647 a[n_] := Sum[Sum[b[n, n, i][[2]] (-1)^(k-i) Binomial[k, i], {i, 0, k}], {k, 0, Max[0, n-2]}];
%t A327647 a /@ Range[0, 27] (* _Jean-François Alcover_, May 03 2020, after Maple *)
%Y A327647 Row sums of A327632, A327648.
%K A327647 nonn
%O A327647 0,4
%A A327647 _Alois P. Heinz_, Sep 20 2019