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A320788
Number of multisets of exactly three partitions of positive integers into distinct parts with total sum of parts equal to n.
2
1, 1, 3, 5, 10, 16, 29, 44, 72, 110, 169, 250, 373, 538, 778, 1104, 1559, 2172, 3016, 4136, 5651, 7653, 10314, 13800, 18389, 24342, 32097, 42096, 54991, 71500, 92637, 119506, 153659, 196831, 251332, 319834, 405824, 513312, 647504, 814448, 1021792, 1278547
OFFSET
3,3
LINKS
FORMULA
a(n) = [x^n y^3] 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, 4)
end:
a:= n-> coeff(b(n$2), x, 3):
seq(a(n), n=3..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, 4}];
a[n_] := SeriesCoefficient[b[n, n], {x, 0, 3}];
a /@ Range[3, 60] (* Jean-François Alcover, Dec 14 2020, after Alois P. Heinz *)
CROSSREFS
Column k=3 of A285229.
Cf. A000009.
Sequence in context: A301653 A253769 A070559 * A000990 A129361 A062773
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Oct 21 2018
STATUS
approved