A339006 Sum over all partitions lambda of n of binomial(|lambda|, |{lambda}|).

1, 1, 3, 5, 11, 20, 40, 72, 130, 227, 395, 671, 1124, 1864, 3040, 4909, 7830, 12394, 19388, 30145, 46395, 70977, 107661, 162383, 243108, 362037, 535684, 788677, 1154605, 1682402, 2439123, 3520706, 5058786, 7239027, 10315920, 14644309, 20709800, 29182353

Offset: 0

Author

Alois P. Heinz, Nov 18 2020

Keywords

• nonn

Comments

|lambda| is the number of parts in lambda and |{lambda}| is the number of distinct parts.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A108492, A339011, A339312.

Programs

Maple

b:= proc(n, i, p, d) option remember; `if`(n=0, binomial(p, d),
     `if`(i<1, 0, add(b(n-i*j, i-1, p+j, `if`(j=0, d, d+1)), j=0..n/i)))
    end:
a:= n-> b(n$2, 0$2):
seq(a(n), n=0..50);

Mathematica

b[n_, i_, p_, d_] := b[n, i, p, d] = If[n == 0, Binomial[p, d],
    If[i<1, 0, Sum[b[n-i*j, i-1, p+j, If[j == 0, d, d+1]], {j, 0, n/i}]]];
a[n_] := b[n, n, 0, 0];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Aug 01 2021, after Alois P. Heinz *)