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URL: https://oeis.org/A327712

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A327712
Sum of multinomials M(n-k; p_1-1, ..., p_k-1), where p = (p_1, ..., p_k) ranges over all compositions of n into distinct parts (k is a composition length).
4
1, 1, 1, 3, 3, 9, 29, 57, 135, 615, 2635, 6273, 25151, 82623, 525281, 2941047, 9100709, 38766777, 205155713, 902705793, 7714938567, 52987356783, 204844103977, 1042657233471, 5520661314689, 38159472253821, 211945677298567, 2404720648663335, 19773733727088813
OFFSET
0,4
COMMENTS
Number of partitions of [n] with distinct block sizes such that each block contains exactly one block size as an element. a(5) = 9: 12345, 1235|4, 124|35, 125|34, 12|345, 134|25, 135|24, 13|245, 1|2345.
MAPLE
with(combinat):
a:= n-> add(multinomial(n-nops(p), map(x-> x-1, p)[], 0), p=map(h->
permute(h)[], select(l-> nops(l)=nops({l[]}), partition(n)))):
seq(a(n), n=0..28);
# Alternative:
a:= proc(m) option remember; local b; b:=
proc(n, i, j) option remember; `if`(i*(i+1)/2>=n,
`if`(n=0, (m-j)!*j!, b(n, i-1, j)+
b(n-i, min(n-i, i-1), j+1)/(i-1)!), 0)
end: b(m$2, 0):
end:
seq(a(n), n=0..28);
MATHEMATICA
a[m_] := a[m] = Module[{b}, b[n_, i_, j_] := b[n, i, j] = If[i(i + 1)/2 >= n, If[n == 0, (m - j)! j!, b[n, i - 1, j] + b[n - i, Min[n - i, i - 1], j + 1]/(i - 1)!], 0]; b[m, m, 0]];
a /@ Range[0, 28] (* Jean-François Alcover, May 10 2020, after 2nd Maple program *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Sep 22 2019
STATUS
approved