A285229 Expansion of g.f. Product_{j>=1} 1/(1-y*x^j)^A000009(j), triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 1, 1, 0, 2, 1, 1, 0, 2, 3, 1, 1, 0, 3, 4, 3, 1, 1, 0, 4, 8, 5, 3, 1, 1, 0, 5, 11, 10, 5, 3, 1, 1, 0, 6, 18, 16, 11, 5, 3, 1, 1, 0, 8, 25, 29, 18, 11, 5, 3, 1, 1, 0, 10, 38, 44, 34, 19, 11, 5, 3, 1, 1, 0, 12, 52, 72, 55, 36, 19, 11, 5, 3, 1, 1

Offset: 0

Author

Alois P. Heinz, Apr 14 2017

Keywords

• nonn
• tabl

Examples

			T(n,k) is the number of multisets of exactly k partitions of positive integers into distinct parts with total sum of parts equal to n.
T(4,1) = 2: {4}, {31}.
T(4,2) = 3: {3,1}, {21,1}, {2,2}.
T(4,3) = 1: {2,1,1}.
T(4,4) = 1: {1,1,1,1}.
Triangle T(n,k) begins:
  1;
  0,  1;
  0,  1,  1;
  0,  2,  1,   1;
  0,  2,  3,   1,  1;
  0,  3,  4,   3,  1,  1;
  0,  4,  8,   5,  3,  1,  1;
  0,  5, 11,  10,  5,  3,  1,  1;
  0,  6, 18,  16, 11,  5,  3,  1,  1;
  0,  8, 25,  29, 18, 11,  5,  3,  1, 1;
  0, 10, 38,  44, 34, 19, 11,  5,  3, 1, 1;
  0, 12, 52,  72, 55, 36, 19, 11,  5, 3, 1, 1;
  0, 15, 75, 110, 96, 60, 37, 19, 11, 5, 3, 1, 1;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..200, flattened
• Index entries for triangles generated by the Multiset Transformation

Crossrefs

Columns k=0..10 give: A000007, A000009 (for n>0), A320787, A320788, A320789, A320790, A320791, A320792, A320793, A320794, A320795.

Row sums give A089259.

T(2n,n) give A285230.

Cf. A061260, A360763.

Programs

Maple

with(numtheory):
g:= proc(n) option remember; `if`(n=0, 1, add(add(
      `if`(d::odd, d, 0), d=divisors(j))*g(n-j), j=1..n)/n)
    end:
b:= proc(n, i) option remember; expand(
      `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))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2)):
seq(T(n), n=0..16);

Mathematica

L[n_] := QPochhammer[x^2]/QPochhammer[x] + O[x]^n;
A[n_] := Module[{c = L[n]}, CoefficientList[#, y]& /@ CoefficientList[ 1/Product[(1 - x^k*y + O[x]^n)^SeriesCoefficient[c, {x, 0, k}], {k, 1, n}], x]];
A[12] // Flatten (* Jean-François Alcover, Jan 19 2020, after Andrew Howroyd *)
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] = 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}]]];
T[n_] := CoefficientList[b[n, n] + O[x]^(n+1), x];
T /@ Range[0, 16] // Flatten (* Jean-François Alcover, Dec 14 2020, after Alois P. Heinz *)

PARI

L(n)={eta(x^2 + O(x*x^n))/eta(x + O(x*x^n))}
A(n)={my(c=L(n), v=Vec(1/prod(k=1, n, (1 - x^k*y + O(x*x^n))^polcoef(c,k)))); vector(#v, n, Vecrev(v[n],n))}
{my(T=A(12)); for(n=1, #T, print(T[n]))} \\ Andrew Howroyd, Dec 29 2019

Formula

G.f.: Product_{j>=1} 1/(1-y*x^j)^A000009(j).