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%I #19 Dec 09 2020 15:12:42
%S 1,1,2,4,14,44,244,1196,9366,62296,584016,5120548,60244028,627389924,
%T 8378159376,106097674780,1652301306958,23655318730276,409987534384504,
%U 6742903763089068,130675390985884516,2396246933608687036,50636625943991790784,1032841246318579471748
%N Number of proper many times partitions of n.
%C In each step at least one part is replaced by the partition of itself into smaller parts. The parts are not resorted.
%H Alois P. Heinz, <a href="/A327644/b327644.txt">Table of n, a(n) for n = 0..400</a>
%H Vaclav Kotesovec, <a href="/A327644/a327644.jpg">Plot of a(n+1)/(n*a(n)) for n = 1..400</a>
%e a(3) = 4: 3, 3->21, 3->111, 3->21->111.
%e a(4) = 14: 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 b:= proc(n, i, k) option remember; `if`(n=0 or k=0, 1, `if`(i>1,
%p b(n, i-1, k), 0) +b(i$2, k-1)*b(n-i, min(n-i, i), k))
%p end:
%p a:= n-> add(add(b(n$2, i)*(-1)^(k-i)*
%p binomial(k, i), i=0..k), k=0..max(0, n-1)):
%p seq(a(n), n=0..23);
%t b[n_, i_, k_] := b[n, i, k] = If[n == 0 || k == 0, 1, If[i > 1, b[n, i - 1, k], 0] + b[i, i, k - 1] b[n - i, Min[n - i, i], k]];
%t a[n_] := Sum[b[n, n, i] (-1)^(k - i) Binomial[k, i], {k, 0, Max[0, n - 1]}, {i, 0, k}];
%t a /@ Range[0, 23] (* _Jean-François Alcover_, Dec 09 2020, after _Alois P. Heinz_ *)
%Y Row sums of A327639.
%Y Cf. A327648.
%K nonn
%O 0,3
%A _Alois P. Heinz_, Sep 20 2019
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