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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.

%I A070289 #66 Oct 05 2024 12:32:42
%S A070289 1,1,2,3,5,7,11,14,20,27,36,47,64,79,102,125,157,193,243,296,366,441,
%T A070289 538,639,773,911,1092,1294,1532,1799,2131,2475,2901,3369,3935,4554,
%U A070289 5292,6084,7033,8087,9292,10617,12198,13880,15874,18039,20541,23263,26414,29838
%N A070289 Number of distinct values of multinomial coefficients ( n / (p1, p2, p3, ...) ) where (p1, p2, p3, ...) runs over all partitions of n.
%H A070289 Alois P. Heinz, <a href="/A070289/b070289.txt">Table of n, a(n) for n = 0..92</a>
%H A070289 George E. Andrews, Arnold Knopfmacher, and Burkhard Zimmermann, <a href="https://doi.org/10.1016/j.jnt.2005.08.012">On the number of distinct multinomial coefficients</a>, Journal of Number Theory 118 (2006), 15-30; <a href="https://arxiv.org/abs/math/0509470">arXiv preprint</a>, arXiv:math/0509470 [math.CO], 2005.
%H A070289 Sergei Viznyuk, <a href="http://phystech.com/ftp/A070289.c">C-Program</a>, <a href="/A070289/a070289_1.c.txt">C-Program, local copy</a>.
%F A070289 a(n) = A215520(n,n) = A215521(2*n,n). - _Alois P. Heinz_, Nov 08 2012
%p A070289 b:= proc(n,i) option remember;
%p A070289       if n=0 then {1} elif i<1 then {} else {b(n, i-1)[],
%p A070289          seq(map(x-> x*i!^j, b(n-i*j, i-1))[], j=1..n/i)} fi
%p A070289     end:
%p A070289 a:= n-> nops(b(n, n)):
%p A070289 seq(a(n), n=0..50);  # _Alois P. Heinz_, Aug 14 2012
%t A070289 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_ *)
%o A070289 (Sage)
%o A070289 def A070289(n):
%o A070289     P = Partitions(n)
%o A070289     M = set(multinomial(list(x)) for x in P)
%o A070289     return len(M)
%o A070289 [A070289(n) for n in range(20)]
%o A070289 # _Joerg Arndt_, Aug 14 2012
%Y A070289 Cf. A000041, A036038, A212855, A215520, A215521, A309951, A309999, A325305, A376661.
%K A070289 nonn
%O A070289 0,3
%A A070289 _Naohiro Nomoto_, May 12 2002
%E A070289 Terms a(n) for n >= 45 corrected by _Joerg Arndt_ and _Alois P. Heinz_, Aug 14 2012