A309999 Number of distinct values of multinomial coefficients M(n;lambda) where lambda ranges over all partitions of n into distinct parts.

1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18, 22, 25, 32, 35, 44, 53, 61, 72, 81, 98, 114, 130, 147, 176, 200, 229, 257, 291, 342, 387, 442, 501, 573, 642, 714, 807, 907, 1037, 1159, 1293, 1458, 1624, 1811, 2024, 2246, 2505, 2785, 3114, 3449, 3795, 4213, 4660

Offset: 0

Author

Alois P. Heinz, Aug 26 2019

Keywords

• nonn

Comments

Differs from A000009 first at n = 15: a(15) = 25 < 27 = A000009(15). There are two repeated multinomial coefficients for n = 15: 1365 = M(15;11,4) = M(15;12,2,1) and 30030 = M(15;9,5,1) = M(15;10,3,2).

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..125
• Wikipedia, Multinomial coefficients
• Wikipedia, Partition (number theory)

Crossrefs

Cf. A000009, A070289.

Programs

Maple

g:= proc(n, i) option remember; `if`(i*(i+1)/2binomial(n, i)*x, g(n-i, min(n-i, i-1)))[], g(n, i-1)[]}))
    end:
a:= n-> nops(g(n$2)):
seq(a(n), n=0..55);

Mathematica

g[n_, i_] := g[n, i] = If[i(i+1)/2 < n, {}, If[n == 0, {1}, Union[ Binomial[n, i] #& /@ g[n - i, Min[n - i, i - 1]], g[n, i - 1]]]];
a[n_] := Length[g[n, n]];
a /@ Range[0, 55] (* Jean-François Alcover, Dec 07 2020, after Alois P. Heinz *)