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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.
2
1, 1, 3, 4, 8, 11, 18, 25, 38, 52, 75, 101, 140, 186, 252, 330, 438, 567, 740, 948, 1221, 1549, 1973, 2482, 3129, 3907, 4884, 6055, 7512, 9255, 11402, 13967, 17102, 20836, 25372, 30760, 37262, 44970, 54221, 65156, 78220, 93622, 111937, 133481, 158996, 188930
OFFSET
2,3
LINKS
FORMULA
a(n) = [x^n y^2] Product_{j>=1} 1/(1-y*x^j)^A000009(j).
MAPLE
g:= proc(n) option remember; `if`(n=0, 1, add(add(`if`(d::odd,
d, 0), d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
end:
b:= proc(n, i) option remember; series(`if`(n=0, 1, `if`(i<1, 0,
add(b(n-i*j, i-1)*x^j*binomial(g(i)+j-1, j), j=0..n/i))), x, 3)
end:
a:= n-> coeff(b(n$2), x, 2):
seq(a(n), n=2..60);
MATHEMATICA
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];
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}];
a[n_] := SeriesCoefficient[b[n, n], {x, 0, 2}];
a /@ Range[2, 60] (* Jean-François Alcover, Dec 14 2020, after Alois P. Heinz *)
CROSSREFS
Column k=2 of A285229.
Cf. A000009.
Sequence in context: A001994 A212544 A349801 * A183151 A288566 A084421
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Oct 21 2018
STATUS
approved