OFFSET
0,3
COMMENTS
Sum of multinomials of number of runs in the partitions of n equals 2^(n-1), so a(n) is less than 2^(n-1).
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
EXAMPLE
The partitions of 4: {4},{3,1},{2,2},{2,1,1},{1,1,1,1} have {1},{1,1},{2},{2,1},{4} runs of equal integers. The sum of the Multinomials of {0},{0,0},{1},{1,0},{3} equals 5.
MAPLE
b:= proc(n, i, p) option remember; `if`(n=0, p!, `if`(i<1, 0,
b(n, i-1, p)+add(b(n-i*j, i-1, p+j-1)/(j-1)!, j=1..n/i)))
end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..45); # Alois P. Heinz, Jul 07 2014
MATHEMATICA
multiplicity2[li:{__Integer}] := (Multinomial@@(-1+Length/@Split[ # ]))&[Sort@li]; Table[Plus@@multiplicity2/@IntegerPartitions[n], {n, 32}]
(* Second program: *)
b[n_, i_, p_] := b[n, i, p] = If[n == 0, p!, If[i < 1, 0, b[n, i - 1, p] + Sum[b[n - i*j, i - 1, p + j - 1]/(j - 1)!, {j, 1, n/i}]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 45}] (* Jean-François Alcover, Aug 29 2016, after Alois P. Heinz *)
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Wouter Meeussen, Mar 22 2003
EXTENSIONS
More terms from Alois P. Heinz, Jul 07 2014
STATUS
approved