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A327648 Number of parts in all proper many times partitions of n.

%I A327648 #15 May 04 2020 12:35:36
%S A327648 0,1,3,9,45,185,1277,7469,67993,514841,5414197,52609653,679432169,
%T A327648 7704502013,111283754969,1515535050805,25257251330321,385282195339393,
%U A327648 7088110874426409,123325149268482781,2520808658222616653,48623257343586890769,1078165538033926164281
%N A327648 Number of parts in all proper many times partitions of n.
%C A327648 In each step at least one part is replaced by the partition of itself into smaller parts. The parts are not resorted.
%H A327648 Alois P. Heinz, <a href="/A327648/b327648.txt">Table of n, a(n) for n = 0..300</a>
%H A327648 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>
%e A327648 a(3) = 9 = 1 + 2 + 3 + 3, counting the (final) parts in: 3, 3->21, 3->111, 3->21->111.
%e A327648 a(4) = 45: 4, 4->31, 4->22, 4->211, 4->1111, 4->31->211, 4->31->1111, 4->22->112, 4->22->211, 4->22->1111, 4->211->1111, 4->31->211->1111, 4->22->112->1111, 4->22->211->1111.
%p A327648 b:= proc(n, i, k) option remember; `if`(n=0, [1, 0],
%p A327648      `if`(k=0, [1, 1], `if`(i<2, 0, b(n, i-1, k))+
%p A327648          (h-> (f-> f +[0, f[1]*h[2]/h[1]])(h[1]*
%p A327648         b(n-i, min(n-i, i), k)))(b(i$2, k-1))))
%p A327648     end:
%p A327648 a:= n-> add(add(b(n$2, i)[2]*(-1)^(k-i)*
%p A327648         binomial(k, i), i=0..k), k=0..n-1):
%p A327648 seq(a(n), n=0..25);
%t A327648 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 A327648 a[n_] := Sum[b[n, n, i][[2]] (-1)^(k - i) Binomial[k, i], {k, 0, n - 1}, {i, 0, k}];
%t A327648 a /@ Range[0, 25] (* _Jean-François Alcover_, May 01 2020, after Maple *)
%Y A327648 Row sums of A327631.
%Y A327648 Cf. A327644, A327647.
%K A327648 nonn
%O A327648 0,3
%A A327648 _Alois P. Heinz_, Sep 20 2019