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A320381 Number of parts in all partitions of 2n with largest multiplicity n.

%I A320381 #13 Jul 16 2023 15:35:12
%S A320381 0,1,5,7,15,18,38,43,81,101,164,206,332,405,613,783,1115,1410,1984,
%T A320381 2483,3402,4281,5697,7147,9417,11702,15167,18861,24093,29782,37745,
%U A320381 46377,58206,71325,88665,108194,133675,162278,199154,241040,293934,354306,429968,516256
%N A320381 Number of parts in all partitions of 2n with largest multiplicity n.
%H A320381 Vaclav Kotesovec, <a href="/A320381/b320381.txt">Table of n, a(n) for n = 0..788</a> (terms 0..400 from Alois P. Heinz)
%F A320381 a(n) = A213177(2n,n).
%e A320381 a(2) = 5 = 3 + 2: [2,1,1], [2,2].
%p A320381 b:= proc(n, i, k) option remember; `if`(n=0, [1, 0], `if`(i<1, 0,
%p A320381       add((l-> [0, l[1]*j]+l)(b(n-i*j, i-1, k)), j=0..min(n/i, k))))
%p A320381     end:
%p A320381 a:= n-> (b(2*n$2, n)-b(2*n$2, n-1))[2]:
%p A320381 seq(a(n), n=0..45);
%t A320381 b[n_, i_, k_] := b[n, i, k] = If[n==0, {1, 0}, If[i<1, {0, 0}, Sum[b[n - i*j, i - 1, k] /. l_List :> {l[[1]], l[[2]] + l[[1]]*j}, {j, 0, Min[n/i, k]}]]];
%t A320381 a[n_] := (b[2n, 2n, n] - b[2n, 2n, n-1])[[2]];
%t A320381 a /@ Range[0, 45] (* _Jean-François Alcover_, Dec 14 2020, after _Alois P. Heinz_ *)
%Y A320381 Cf. A213177.
%K A320381 nonn
%O A320381 0,3
%A A320381 _Alois P. Heinz_, Oct 11 2018