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A070289
Number of distinct values of multinomial coefficients ( n / (p1, p2, p3, ...) ) where (p1, p2, p3, ...) runs over all partitions of n.
20
1, 1, 2, 3, 5, 7, 11, 14, 20, 27, 36, 47, 64, 79, 102, 125, 157, 193, 243, 296, 366, 441, 538, 639, 773, 911, 1092, 1294, 1532, 1799, 2131, 2475, 2901, 3369, 3935, 4554, 5292, 6084, 7033, 8087, 9292, 10617, 12198, 13880, 15874, 18039, 20541, 23263, 26414, 29838
OFFSET
0,3
LINKS
George E. Andrews, Arnold Knopfmacher, and Burkhard Zimmermann, On the number of distinct multinomial coefficients, Journal of Number Theory 118 (2006), 15-30; arXiv preprint, arXiv:math/0509470 [math.CO], 2005.
FORMULA
a(n) = A215520(n,n) = A215521(2*n,n). - Alois P. Heinz, Nov 08 2012
MAPLE
b:= proc(n, i) option remember;
if n=0 then {1} elif i<1 then {} else {b(n, i-1)[],
seq(map(x-> x*i!^j, b(n-i*j, i-1))[], j=1..n/i)} fi
end:
a:= n-> nops(b(n, n)):
seq(a(n), n=0..50); # Alois P. Heinz, Aug 14 2012
MATHEMATICA
b[n_, i_] := b[n, i] = If[n == 0, {1}, If[i<1, {}, Union[Join[b[n, i-1], Flatten[ Table[Function[{x}, x*i!^j] /@ b[n-i*j, i-1], {j, 1, n/i}]]]]]]; a[n_] := Length[b[n, n]]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Mar 23 2015, after Alois P. Heinz *)
PROG
(Sage)
def A070289(n):
P = Partitions(n)
M = set(multinomial(list(x)) for x in P)
return len(M)
[A070289(n) for n in range(20)]
# Joerg Arndt, Aug 14 2012
KEYWORD
nonn
AUTHOR
Naohiro Nomoto, May 12 2002
EXTENSIONS
Terms a(n) for n >= 45 corrected by Joerg Arndt and Alois P. Heinz, Aug 14 2012
STATUS
approved