A320373 - OEIS

%I #11 Dec 13 2020 09:31:21

%S 3,0,4,7,13,14,27,33,55,72,107,137,196,250,344,442,588,750,982,1234,

%T 1591,1992,2523,3135,3944,4857,6035,7408,9121,11109,13599,16465,20004,

%U 24122,29112,34927,41952,50078,59836,71169,84625,100219,118716,140061,165225

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

%H Alois P. Heinz, <a href="/A320373/b320373.txt">Table of n, a(n) for n = 3..5000</a>

%F a(n) ~ log(2) * exp(Pi*sqrt(n/2)) / (Pi * 2^(1/4) * 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])(3):

%p seq(a(n), n=3..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 = 3}, (b[n, n, k] - b[n, n, k - 1])[[2]]];

%t a /@ Range[3, 60] (* _Jean-François Alcover_, Dec 13 2020, after _Alois P. Heinz_ *)

%Y Column k=3 of A213177.

%K nonn

%O 3,1

%A _Alois P. Heinz_, Oct 11 2018