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Sum over all partitions lambda of n of binomial(|lambda|, |{lambda}|).
14

%I #15 Aug 01 2021 13:17:07

%S 1,1,3,5,11,20,40,72,130,227,395,671,1124,1864,3040,4909,7830,12394,

%T 19388,30145,46395,70977,107661,162383,243108,362037,535684,788677,

%U 1154605,1682402,2439123,3520706,5058786,7239027,10315920,14644309,20709800,29182353

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

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

%H Alois P. Heinz, <a href="/A339006/b339006.txt">Table of n, a(n) for n = 0..1000</a>

%p b:= proc(n, i, p, d) option remember; `if`(n=0, binomial(p, d),

%p `if`(i<1, 0, add(b(n-i*j, i-1, p+j, `if`(j=0, d, d+1)), j=0..n/i)))

%p end:

%p a:= n-> b(n$2, 0$2):

%p seq(a(n), n=0..50);

%t b[n_, i_, p_, d_] := b[n, i, p, d] = If[n == 0, Binomial[p, d],

%t If[i<1, 0, Sum[b[n-i*j, i-1, p+j, If[j == 0, d, d+1]], {j, 0, n/i}]]];

%t a[n_] := b[n, n, 0, 0];

%t Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Aug 01 2021, after _Alois P. Heinz_ *)

%Y Cf. A108492, A339011, A339312.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Nov 18 2020