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A350175 Sum of the distinct block sizes over all partitions of [n].

%I A350175 #25 Jan 11 2022 10:46:05
%S A350175 0,1,3,13,45,196,888,4383,22879,129163,768913,4849912,32202712,
%T A350175 224672241,1640679589,12517008985,99484656169,822410210044,
%U A350175 7055883373604,62730142658947,576984726864147,5482889832932123,53757450049841167,543169144098559606,5649499728403949184
%N A350175 Sum of the distinct block sizes over all partitions of [n].
%H A350175 Alois P. Heinz, <a href="/A350175/b350175.txt">Table of n, a(n) for n = 0..400</a>
%H A350175 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A350175 a(n) mod 2 = A131719(n).
%e A350175 a(3) = 13 = 1*3 + 3*(1+2) + 1: 123, 1|23, 13|2, 12|3, 1|2|3.
%p A350175 b:= proc(n, i, c) option remember; `if`(n=0, c,
%p A350175       `if`(i<1, 0, add(b(n-j*i, i-1, c+i*signum(j))*
%p A350175       combinat[multinomial](n, n-i*j, i$j)/j!, j=0..n/i)))
%p A350175     end:
%p A350175 a:= n-> b(n$2, 0):
%p A350175 seq(a(n), n=0..30);
%t A350175 multinomial[n_, k_List] := n!/Times @@ (k!);
%t A350175 b[n_, i_, c_] := b[n, i, c] = If[n == 0, c,
%t A350175      If[i < 1, 0, Sum[b[n - j*i, i - 1, c + i*Sign[j]]*
%t A350175      multinomial[n, Join[{n - i*j}, Table[i, {j}]]]/j!, {j, 0, n/i}]]];
%t A350175 a[n_] := b[n, n, 0];
%t A350175 Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Jan 11 2022, after _Alois P. Heinz_ *)
%Y A350175 Cf. A000110, A070071, A131719, A132963.
%K A350175 nonn
%O A350175 0,3
%A A350175 _Alois P. Heinz_, Jan 06 2022