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Sum of the distinct block sizes over all partitions of [n].
2

%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