OFFSET
0,5
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows n = 0..50)
FORMULA
G.f.: Product_{j>=1} (1 + y*x^j)^A000041(j). - Andrew Howroyd, Dec 29 2019
EXAMPLE
Triangle begins:
1
0 1
0 2 0
0 3 2 0
0 5 4 0 0
0 7 11 1 0 0
0 11 20 6 0 0 0
0 15 40 16 0 0 0 0
0 22 68 40 3 0 0 0 0
...
Row n = 5 counts the following sets of multisets:
{{5}} {{1},{4}} {{1},{2},{1,1}}
{{1,4}} {{2},{3}}
{{2,3}} {{1},{1,3}}
{{1,1,3}} {{1},{2,2}}
{{1,2,2}} {{2},{1,2}}
{{1,1,1,2}} {{3},{1,1}}
{{1,1,1,1,1}} {{1},{1,1,2}}
{{1,1},{1,2}}
{{2},{1,1,1}}
{{1},{1,1,1,1}}
{{1,1},{1,1,1}}
MAPLE
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(binomial(
combinat[numbpart](i), j)*expand(b(n-i*j, i-1)*x^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..14); # Alois P. Heinz, Dec 30 2019
MATHEMATICA
ppl[n_, k_]:=Switch[k, 0, {n}, 1, IntegerPartitions[n], _, Join@@Table[Union[Sort/@Tuples[ppl[#, k-1]&/@ptn]], {ptn, IntegerPartitions[n]}]];
Table[Length[Select[ppl[n, 2], And[UnsameQ@@#, Length[#]==k]&]], {n, 0, 10}, {k, 0, n}]
(* Second program: *)
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[Binomial[
PartitionsP[i], j]*Expand[b[n - i*j, i - 1]*x^j], {j, 0, n/i}]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, n]];
T /@ Range[0, 14] // Flatten (* Jean-François Alcover, May 18 2021, after Alois P. Heinz *)
PROG
(PARI)
A(n)={my(v=Vec(prod(k=1, n, (1 + x^k*y + O(x*x^n))^numbpart(k)))); vector(#v, n, Vecrev(v[n], n))}
{my(T=A(12)); for(n=1, #T, print(T[n]))} \\ Andrew Howroyd, Dec 29 2019
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Dec 19 2019
STATUS
approved