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A372632 Sum over all partitions of n of the number of elements with minimal multiplicity in their partition.

%I A372632 #16 May 10 2024 08:49:56
%S A372632 0,1,2,4,6,9,16,21,33,47,67,90,134,172,242,321,434,558,761,961,1279,
%T A372632 1627,2112,2657,3452,4292,5481,6824,8619,10634,13381,16389,20425,
%U A372632 24989,30864,37556,46221,55912,68337,82510,100262,120476,145855,174562,210272,251065
%N A372632 Sum over all partitions of n of the number of elements with minimal multiplicity in their partition.
%H A372632 Alois P. Heinz, <a href="/A372632/b372632.txt">Table of n, a(n) for n = 0..2000</a>
%F A372632 a(n) = Sum_{k=0..A003056(n)} k * A362615(n,k).
%p A372632 b:= proc(n, i, m, t) option remember; `if`(n=0, t,
%p A372632       `if`(i<1, 0, b(n, i-1, m, t)+add(b(n-i*j, i-1, min(j, m),
%p A372632       `if`(j<m, 1, `if`(j=m, t+1, t))), j=1..n/i)))
%p A372632     end:
%p A372632 a:= n-> b(n$3, 0):
%p A372632 seq(a(n), n=0..45);
%t A372632 b[n_, i_, m_, t_] := b[n, i, m, t] = If[n == 0, t,
%t A372632    If[i < 1, 0, b[n, i - 1, m, t] + Sum[b[n - i*j, i - 1, Min[j, m],
%t A372632    If[j < m, 1, If[j == m, t + 1, t]]], {j, 1, n/i}]]];
%t A372632 a[n_] := b[n, n, n, 0];
%t A372632 Table[a[n], {n, 0, 45}] (* _Jean-François Alcover_, May 10 2024, after _Alois P. Heinz_ *)
%Y A372632 Cf. A000041, A003056, A362615, A372542.
%K A372632 nonn
%O A372632 0,3
%A A372632 _Alois P. Heinz_, May 07 2024