A320788 Number of multisets of exactly three partitions of positive integers into distinct parts with total sum of parts equal to n.

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

Alois P. Heinz, Table of n, a(n) for n = 3..1000

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

Adjacent sequences: A320785 A320786 A320787 * A320789 A320790 A320791

Keyword

nonn

Author

Alois P. Heinz, Oct 21 2018

Status

approved