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A320789 Number of multisets of exactly four partitions of positive integers into distinct parts with total sum of parts equal to n.

%I A320789 #7 Dec 14 2020 05:14:01
%S A320789 1,1,3,5,11,18,34,55,96,152,248,386,607,921,1405,2092,3112,4551,6635,
%T A320789 9545,13683,19401,27393,38346,53441,73928,101840,139398,190020,257601,
%U A320789 347836,467381,625686,833917,1107547,1465136,1931754,2537747,3323490,4338012,5645645
%N A320789 Number of multisets of exactly four partitions of positive integers into distinct parts with total sum of parts equal to n.
%H A320789 Alois P. Heinz, <a href="/A320789/b320789.txt">Table of n, a(n) for n = 4..1000</a>
%F A320789 a(n) = [x^n y^4] Product_{j>=1} 1/(1-y*x^j)^A000009(j).
%p A320789 g:= proc(n) option remember; `if`(n=0, 1, add(add(`if`(d::odd,
%p A320789       d, 0), d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
%p A320789     end:
%p A320789 b:= proc(n, i) option remember; series(`if`(n=0, 1, `if`(i<1, 0,
%p A320789       add(b(n-i*j, i-1)*x^j*binomial(g(i)+j-1, j), j=0..n/i))), x, 5)
%p A320789     end:
%p A320789 a:= n-> coeff(b(n$2), x, 4):
%p A320789 seq(a(n), n=4..60);
%t A320789 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 A320789 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, 5}];
%t A320789 a[n_] := SeriesCoefficient[b[n, n], {x, 0, 4}];
%t A320789 a /@ Range[4, 60] (* _Jean-François Alcover_, Dec 14 2020, after _Alois P. Heinz_ *)
%Y A320789 Column k=4 of A285229.
%Y A320789 Cf. A000009.
%K A320789 nonn
%O A320789 4,3
%A A320789 _Alois P. Heinz_, Oct 21 2018