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%I #17 Dec 18 2020 04:01:16
%S 0,0,1,4,25,108,788,4740,44445,339632,3625136,35508536,462626736,
%T 5273725108,76634997096,1047347436984,17542238923677,268193251446228,
%U 4949536256552648,86303019303031400,1768833677916545596,34165810747993948664,759192269597947084836
%N Total number of steps in all 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="/A327646/b327646.txt">Table of n, a(n) for n = 0..400</a>
%F a(n) = Sum_{k=1..n-1} k * A327639(n,k).
%e a(3) = 4 = 0+1+1+2, counting steps "->" in: 3, 3->21, 3->111, 3->21->111.
%e a(4) = 25: 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(k*add(b(n$2, i)*(-1)^(k-i)*
%p binomial(k, i), i=0..k), k=1..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[k Sum[b[n, n, i] (-1)^(k - i) Binomial[k, i], {i, 0, k}], {k, 1, n - 1}];
%t a /@ Range[0, 23] (* _Jean-François Alcover_, Dec 18 2020, after _Alois P. Heinz_ *)
%Y Cf. A327639.
%K nonn
%O 0,4
%A _Alois P. Heinz_, Sep 20 2019
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