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%I #8 Dec 14 2020 05:13:54
%S 1,1,3,5,10,16,29,44,72,110,169,250,373,538,778,1104,1559,2172,3016,
%T 4136,5651,7653,10314,13800,18389,24342,32097,42096,54991,71500,92637,
%U 119506,153659,196831,251332,319834,405824,513312,647504,814448,1021792,1278547
%N Number of multisets of exactly three partitions of positive integers into distinct parts with total sum of parts equal to n.
%H Alois P. Heinz, <a href="/A320788/b320788.txt">Table of n, a(n) for n = 3..1000</a>
%F a(n) = [x^n y^3] Product_{j>=1} 1/(1-y*x^j)^A000009(j).
%p g:= proc(n) option remember; `if`(n=0, 1, add(add(`if`(d::odd,
%p d, 0), d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
%p end:
%p b:= proc(n, i) option remember; series(`if`(n=0, 1, `if`(i<1, 0,
%p add(b(n-i*j, i-1)*x^j*binomial(g(i)+j-1, j), j=0..n/i))), x, 4)
%p end:
%p a:= n-> coeff(b(n$2), x, 3):
%p seq(a(n), n=3..60);
%t g[n_] := g[n] = If[n == 0, 1, Sum[Sum[If[OddQ[d], d, 0], {d, Divisors[j]}]* g[n - j], {j, 1, n}]/n];
%t b[n_, i_] := b[n, i] = Series[If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i - 1]*x^j*Binomial[g[i] + j - 1, j], {j, 0, n/i}]]], {x, 0, 4}];
%t a[n_] := SeriesCoefficient[b[n, n], {x, 0, 3}];
%t a /@ Range[3, 60] (* _Jean-François Alcover_, Dec 14 2020, after _Alois P. Heinz_ *)
%Y Column k=3 of A285229.
%Y Cf. A000009.
%K nonn
%O 3,3
%A _Alois P. Heinz_, Oct 21 2018
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