A320787 Number of multisets of exactly two partitions of positive integers into distinct parts with total sum of parts equal to n.

1, 1, 3, 4, 8, 11, 18, 25, 38, 52, 75, 101, 140, 186, 252, 330, 438, 567, 740, 948, 1221, 1549, 1973, 2482, 3129, 3907, 4884, 6055, 7512, 9255, 11402, 13967, 17102, 20836, 25372, 30760, 37262, 44970, 54221, 65156, 78220, 93622, 111937, 133481, 158996, 188930

Offset

2,3

Links

Alois P. Heinz, Table of n, a(n) for n = 2..1000

Formula

a(n) = [x^n y^2] 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, 3)
    end:
a:= n-> coeff(b(n$2), x, 2):
seq(a(n), n=2..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, 3}];
a[n_] := SeriesCoefficient[b[n, n], {x, 0, 2}];
a /@ Range[2, 60] (* Jean-François Alcover, Dec 14 2020, after Alois P. Heinz *)

Crossrefs

Column k=2 of A285229.

Cf. A000009.

Sequence in context: A001994 A212544 A349801 * A183151 A288566 A084421

Adjacent sequences: A320784 A320785 A320786 * A320788 A320789 A320790

Keyword

nonn

Author

Alois P. Heinz, Oct 21 2018

Status

approved