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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.
15
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
0,8
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
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.
Sequence in context: A123340 A360455 A267486 * A227425 A333213 A301636
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Apr 14 2017
STATUS
approved