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

0,4

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)

Maple

g:= proc(n, i) option remember; `if`(i*(i+1)/2<n, {}, `if`(n=0, {1},
     {map(x->binomial(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 *)

Crossrefs

Cf. A000009, A070289.

Sequence in context: A027581 A058706 A034143 * A034144 A347586 A287997

Adjacent sequences: A309996 A309997 A309998 * A310000 A310001 A310002

Keyword

nonn

Author

Alois P. Heinz, Aug 26 2019

Status

approved