OFFSET
0,8
LINKS
FORMULA
G.f.: Product_{j>=1} 1/(1-y*x^j)^A000009(j).
EXAMPLE
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;
...
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Apr 14 2017
STATUS
approved