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

5,3

Links

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

Formula

a(n) = [x^n y^5] 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, 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 *)

Crossrefs

Column k=5 of A285229.

Cf. A000009.

Sequence in context: A329178 A298348 A303005 * A175783 A078722 A320791

Adjacent sequences: A320787 A320788 A320789 * A320791 A320792 A320793

Keyword

nonn

Author

Alois P. Heinz, Oct 21 2018

Status

approved