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A372542 Total number of modes in all partitions of n.

%I A372542 #31 May 07 2024 18:53:34
%S A372542 0,1,2,4,6,9,16,20,30,42,61,76,112,138,189,248,325,407,539,667,865,
%T A372542 1083,1369,1690,2140,2624,3268,4009,4954,6022,7417,8968,10946,13218,
%U A372542 16023,19256,23264,27819,33415,39873,47682,56654,67527,79962,94909,112130,132578
%N A372542 Total number of modes in all partitions of n.
%C A372542 Each element of a partition with maximal multiplicity is a mode of this partition.
%H A372542 Alois P. Heinz, <a href="/A372542/b372542.txt">Table of n, a(n) for n = 0..2000</a>
%F A372542 a(n) = Sum_{k=0..A003056(n)} k * A362614(n,k).
%e A372542 a(5) = 9 = 1 + 2 + 2 + 1 + 1 + 1 + 1: 5, 32, 41, 221, 311, 2111, 11111.
%p A372542 b:= proc(n, i, m, t) option remember; `if`(n=0, t, `if`(i<1, 0, add(
%p A372542       b(n-i*j, i-1, max(j, m), `if`(j>m, 1, `if`(j=m, t+1, t))), j=0..n/i)))
%p A372542     end:
%p A372542 a:= n-> b(n$2, 0$2):
%p A372542 seq(a(n), n=0..46);
%Y A372542 Cf. A000041, A003056, A362614, A372632.
%K A372542 nonn
%O A372542 0,3
%A A372542 _Alois P. Heinz_, May 05 2024