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A320787 Number of multisets of exactly two partitions of positive integers into distinct parts with total sum of parts equal to n.

%I A320787 #9 Dec 14 2020 05:13:48
%S A320787 1,1,3,4,8,11,18,25,38,52,75,101,140,186,252,330,438,567,740,948,1221,
%T A320787 1549,1973,2482,3129,3907,4884,6055,7512,9255,11402,13967,17102,20836,
%U A320787 25372,30760,37262,44970,54221,65156,78220,93622,111937,133481,158996,188930
%N A320787 Number of multisets of exactly two partitions of positive integers into distinct parts with total sum of parts equal to n.
%H A320787 Alois P. Heinz, <a href="/A320787/b320787.txt">Table of n, a(n) for n = 2..1000</a>
%F A320787 a(n) = [x^n y^2] Product_{j>=1} 1/(1-y*x^j)^A000009(j).
%p A320787 g:= proc(n) option remember; `if`(n=0, 1, add(add(`if`(d::odd,
%p A320787       d, 0), d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
%p A320787     end:
%p A320787 b:= proc(n, i) option remember; series(`if`(n=0, 1, `if`(i<1, 0,
%p A320787       add(b(n-i*j, i-1)*x^j*binomial(g(i)+j-1, j), j=0..n/i))), x, 3)
%p A320787     end:
%p A320787 a:= n-> coeff(b(n$2), x, 2):
%p A320787 seq(a(n), n=2..60);
%t A320787 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 A320787 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, 3}];
%t A320787 a[n_] := SeriesCoefficient[b[n, n], {x, 0, 2}];
%t A320787 a /@ Range[2, 60] (* _Jean-François Alcover_, Dec 14 2020, after _Alois P. Heinz_ *)
%Y A320787 Column k=2 of A285229.
%Y A320787 Cf. A000009.
%K A320787 nonn
%O A320787 2,3
%A A320787 _Alois P. Heinz_, Oct 21 2018