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A320790 Number of multisets of exactly five partitions of positive integers into distinct parts with total sum of parts equal to n.

%I A320790 #7 Dec 14 2020 05:14:08
%S A320790 1,1,3,5,11,19,36,60,107,176,296,475,770,1211,1906,2939,4518,6842,
%T A320790 10313,15363,22770,33424,48802,70688,101854,145755,207528,293704,
%U A320790 413691,579571,808328,1121923,1550645,2133751,2924579,3992307,5429751,7357195,9934357
%N A320790 Number of multisets of exactly five partitions of positive integers into distinct parts with total sum of parts equal to n.
%H A320790 Alois P. Heinz, <a href="/A320790/b320790.txt">Table of n, a(n) for n = 5..1000</a>
%F A320790 a(n) = [x^n y^5] Product_{j>=1} 1/(1-y*x^j)^A000009(j).
%p A320790 g:= proc(n) option remember; `if`(n=0, 1, add(add(`if`(d::odd,
%p A320790       d, 0), d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
%p A320790     end:
%p A320790 b:= proc(n, i) option remember; series(`if`(n=0, 1, `if`(i<1, 0,
%p A320790       add(b(n-i*j, i-1)*x^j*binomial(g(i)+j-1, j), j=0..n/i))), x, 6)
%p A320790     end:
%p A320790 a:= n-> coeff(b(n$2), x, 5):
%p A320790 seq(a(n), n=5..60);
%t A320790 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 A320790 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, 6}];
%t A320790 a[n_] := SeriesCoefficient[b[n, n], {x, 0, 5}];
%t A320790 a /@ Range[5, 60] (* _Jean-François Alcover_, Dec 14 2020, after _Alois P. Heinz_ *)
%Y A320790 Column k=5 of A285229.
%Y A320790 Cf. A000009.
%K A320790 nonn
%O A320790 5,3
%A A320790 _Alois P. Heinz_, Oct 21 2018