A320789 Number of multisets of exactly four partitions of positive integers into distinct parts with total sum of parts equal to n.

1, 1, 3, 5, 11, 18, 34, 55, 96, 152, 248, 386, 607, 921, 1405, 2092, 3112, 4551, 6635, 9545, 13683, 19401, 27393, 38346, 53441, 73928, 101840, 139398, 190020, 257601, 347836, 467381, 625686, 833917, 1107547, 1465136, 1931754, 2537747, 3323490, 4338012, 5645645

Offset

4,3

Links

Alois P. Heinz, Table of n, a(n) for n = 4..1000

Formula

a(n) = [x^n y^4] 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, 5)
    end:
a:= n-> coeff(b(n$2), x, 4):
seq(a(n), n=4..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, 5}];
a[n_] := SeriesCoefficient[b[n, n], {x, 0, 4}];
a /@ Range[4, 60] (* Jean-François Alcover, Dec 14 2020, after Alois P. Heinz *)

Crossrefs

Column k=4 of A285229.

Cf. A000009.

Sequence in context: A244473 A023597 A281357 * A269628 A162891 A320351

Adjacent sequences: A320786 A320787 A320788 * A320790 A320791 A320792

Keyword

nonn

Author

Alois P. Heinz, Oct 21 2018

Status

approved