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