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A330463 Triangle read by rows where T(n,k) is the number of k-element sets of nonempty multisets of positive integers with total sum n. 8
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, 0, 30, 120, 91, 11, 0, 0, 0, 0, 0, 0, 42, 195, 186, 41, 0, 0, 0, 0, 0, 0, 0, 56, 320, 367, 105, 3, 0, 0, 0, 0, 0, 0 (list; table; graph; refs; listen; history; text; internal format)
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
CROSSREFS
Row sums are A261049.
Column k = 1 is A000041.
Multisets of multisets are A061260, with row sums A001970.
Sets of sets are A330462, with row sums A050342.
Multisets of sets are A285229, with row sums A089259.
Sets of disjoint sets are A330460, with row sums A294617.
Sequence in context: A143161 A225853 A342128 * A142886 A099026 A341410
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Dec 19 2019
STATUS
approved

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Last modified March 18 22:56 EDT 2024. Contains 370952 sequences. (Running on oeis4.)