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A330765 Total number of blocks in all set partitions of strict integer partitions of n.

%I A330765 #14 May 08 2020 16:34:31
%S A330765 0,1,1,4,4,7,17,20,30,43,90,103,160,210,304,515,646,894,1223,1659,
%T A330765 2176,3484,4226,5873,7638,10335,13150,17695,24974,31394,41383,53766,
%U A330765 69718,89573,115613,146344,201625,247880,322099,406445,524634,654298,839584,1043012
%N A330765 Total number of blocks in all set partitions of strict integer partitions of n.
%H A330765 Alois P. Heinz, <a href="/A330765/b330765.txt">Table of n, a(n) for n = 0..5000</a>
%H A330765 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A330765 a(n) = Sum_{k=1..A003056(n)} k * A330460(n,k).
%F A330765 a(n) = Sum_{k=1..A003056(n)} k * A330759(n,k).
%p A330765 b:= proc(n, i, k) option remember; `if`(i*(i+1)/2<n, 0,
%p A330765       `if`(n=0, k, b(n, i-1, k)+(t-> b(n-i, t, k)*k
%p A330765         +b(n-i, t, k+1))(min(n-i, i-1))))
%p A330765     end:
%p A330765 a:= n-> b(n$2, 0):
%p A330765 seq(a(n), n=0..50);
%t A330765 b[n_, i_, k_] := b[n, i, k] = If[i(i+1)/2 < n, 0, If[n==0, k, b[n, i-1, k] + b[n-i, #, k] k + b[n-i, #, k+1]&[Min[n-i, i-1]]]];
%t A330765 a[n_] := b[n, n, 0];
%t A330765 a /@ Range[0, 50] (* _Jean-François Alcover_, May 08 2020, after Maple *)
%Y A330765 Cf. A003056, A330460, A330759.
%K A330765 nonn
%O A330765 0,4
%A A330765 _Alois P. Heinz_, Dec 29 2019