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

1, 1, 3, 5, 11, 19, 36, 60, 107, 176, 296, 475, 770, 1211, 1906, 2939, 4518, 6842, 10313, 15363, 22770, 33424, 48802, 70688, 101854, 145755, 207528, 293704, 413691, 579571, 808328, 1121923, 1550645, 2133751, 2924579, 3992307, 5429751, 7357195, 9934357

Offset: 5

Author

Alois P. Heinz, Oct 21 2018

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 5..1000

Crossrefs

Column k=5 of A285229.

Cf. A000009.

Programs

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, 6)
    end:
a:= n-> coeff(b(n$2), x, 5):
seq(a(n), n=5..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, 6}];
a[n_] := SeriesCoefficient[b[n, n], {x, 0, 5}];
a /@ Range[5, 60] (* Jean-François Alcover, Dec 14 2020, after Alois P. Heinz *)

Formula

a(n) = [x^n y^5] Product_{j>=1} 1/(1-y*x^j)^A000009(j).