A320372 - OEIS

%I #11 Dec 13 2020 09:31:11

%S 2,0,5,6,9,13,23,30,44,58,85,108,149,191,258,326,425,532,688,852,1082,

%T 1331,1670,2042,2531,3068,3771,4554,5543,6653,8051,9607,11543,13722,

%U 16377,19390,23023,27132,32073,37660,44303,51834,60744,70813,82666,96082,111759

%N Number of parts in all partitions of n with largest multiplicity two.

%H Alois P. Heinz, <a href="/A320372/b320372.txt">Table of n, a(n) for n = 2..5000</a>

%F a(n) ~ log(3) * exp(2*Pi*sqrt(n)/3) / (2*Pi*n^(1/4)). - _Vaclav Kotesovec_, Oct 25 2018

%p b:= proc(n, i, k) option remember; `if`(n=0, [1, 0], `if`(i<1, 0,

%p       add((l-> [0, l[1]*j]+l)(b(n-i*j, i-1, k)), j=0..min(n/i, k))))

%p     end:

%p a:= n-> (k-> (b(n$2, k)-b(n$2, k-1))[2])(2):

%p seq(a(n), n=2..60);

%t b[n_, i_, k_] := b[n, i, k] = If[n==0, {1, 0}, If[i<1, {0, 0}, Sum[ Function[l, {0, l[[1]]*j} + l][b[n - i j, i-1, k]], {j, 0, Min[n/i, k]}]]];

%t a[n_] := With[{k = 2}, (b[n, n, k] - b[n, n, k - 1])[[2]]];

%t a /@ Range[2, 60] (* _Jean-François Alcover_, Dec 13 2020, after _Alois P. Heinz_ *)

%Y Column k=2 of A213177.

%K nonn

%O 2,1

%A _Alois P. Heinz_, Oct 11 2018