A070289 Number of distinct values of multinomial coefficients ( n / (p1, p2, p3, ...) ) where (p1, p2, p3, ...) runs over all partitions of n.

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

Author

Naohiro Nomoto, May 12 2002

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..92
• 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.
• Sergei Viznyuk, C-Program, C-Program, local copy.

Crossrefs

Cf. A000041, A036038, A212855, A215520, A215521, A309951, A309999, A325305, A376661.

Programs

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 *)

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

Formula

a(n) = A215520(n,n) = A215521(2*n,n). - Alois P. Heinz, Nov 08 2012

Extensions

Terms a(n) for n >= 45 corrected by Joerg Arndt and Alois P. Heinz, Aug 14 2012