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A320373 Number of parts in all partitions of n with largest multiplicity three.

%I A320373 #11 Dec 13 2020 09:31:21
%S A320373 3,0,4,7,13,14,27,33,55,72,107,137,196,250,344,442,588,750,982,1234,
%T A320373 1591,1992,2523,3135,3944,4857,6035,7408,9121,11109,13599,16465,20004,
%U A320373 24122,29112,34927,41952,50078,59836,71169,84625,100219,118716,140061,165225
%N A320373 Number of parts in all partitions of n with largest multiplicity three.
%H A320373 Alois P. Heinz, <a href="/A320373/b320373.txt">Table of n, a(n) for n = 3..5000</a>
%F A320373 a(n) ~ log(2) * exp(Pi*sqrt(n/2)) / (Pi * 2^(1/4) * n^(1/4)). - _Vaclav Kotesovec_, Oct 25 2018
%p A320373 b:= proc(n, i, k) option remember; `if`(n=0, [1, 0], `if`(i<1, 0,
%p A320373       add((l-> [0, l[1]*j]+l)(b(n-i*j, i-1, k)), j=0..min(n/i, k))))
%p A320373     end:
%p A320373 a:= n-> (k-> (b(n$2, k)-b(n$2, k-1))[2])(3):
%p A320373 seq(a(n), n=3..60);
%t A320373 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 A320373 a[n_] := With[{k = 3}, (b[n, n, k] - b[n, n, k - 1])[[2]]];
%t A320373 a /@ Range[3, 60] (* _Jean-François Alcover_, Dec 13 2020, after _Alois P. Heinz_ *)
%Y A320373 Column k=3 of A213177.
%K A320373 nonn
%O A320373 3,1
%A A320373 _Alois P. Heinz_, Oct 11 2018