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

%I A339006 #15 Aug 01 2021 13:17:07
%S A339006 1,1,3,5,11,20,40,72,130,227,395,671,1124,1864,3040,4909,7830,12394,
%T A339006 19388,30145,46395,70977,107661,162383,243108,362037,535684,788677,
%U A339006 1154605,1682402,2439123,3520706,5058786,7239027,10315920,14644309,20709800,29182353
%N A339006 Sum over all partitions lambda of n of binomial(|lambda|, |{lambda}|).
%C A339006 |lambda| is the number of parts in lambda and |{lambda}| is the number of distinct parts.
%H A339006 Alois P. Heinz, <a href="/A339006/b339006.txt">Table of n, a(n) for n = 0..1000</a>
%p A339006 b:= proc(n, i, p, d) option remember; `if`(n=0, binomial(p, d),
%p A339006      `if`(i<1, 0, add(b(n-i*j, i-1, p+j, `if`(j=0, d, d+1)), j=0..n/i)))
%p A339006     end:
%p A339006 a:= n-> b(n$2, 0$2):
%p A339006 seq(a(n), n=0..50);
%t A339006 b[n_, i_, p_, d_] := b[n, i, p, d] = If[n == 0, Binomial[p, d],
%t A339006     If[i<1, 0, Sum[b[n-i*j, i-1, p+j, If[j == 0, d, d+1]], {j, 0, n/i}]]];
%t A339006 a[n_] := b[n, n, 0, 0];
%t A339006 Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Aug 01 2021, after _Alois P. Heinz_ *)
%Y A339006 Cf. A108492, A339011, A339312.
%K A339006 nonn
%O A339006 0,3
%A A339006 _Alois P. Heinz_, Nov 18 2020