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A330462 Triangle read by rows where T(n,k) is the number of k-element sets of nonempty sets of positive integers with total sum n. 9
1, 0, 1, 0, 1, 0, 0, 2, 1, 0, 0, 2, 2, 0, 0, 0, 3, 4, 0, 0, 0, 0, 4, 6, 2, 0, 0, 0, 0, 5, 11, 3, 0, 0, 0, 0, 0, 6, 16, 8, 0, 0, 0, 0, 0, 0, 8, 25, 15, 1, 0, 0, 0, 0, 0, 0, 10, 35, 28, 4, 0, 0, 0, 0, 0, 0, 0, 12, 52, 46, 9, 0, 0, 0, 0, 0, 0, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,8
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)^A000009(j). - Andrew Howroyd, Dec 29 2019
EXAMPLE
Triangle begins:
1
0 1
0 1 0
0 2 1 0
0 2 2 0 0
0 3 4 0 0 0
0 4 6 2 0 0 0
0 5 11 3 0 0 0 0
0 6 16 8 0 0 0 0 0
0 8 25 15 1 0 0 0 0 0
0 10 35 28 4 0 0 0 0 0 0
...
Row n = 7 counts the following set-systems:
{{7}} {{1},{6}} {{1},{2},{4}}
{{1,6}} {{2},{5}} {{1},{2},{1,3}}
{{2,5}} {{3},{4}} {{1},{3},{1,2}}
{{3,4}} {{1},{1,5}}
{{1,2,4}} {{1},{2,4}}
{{2},{1,4}}
{{2},{2,3}}
{{3},{1,3}}
{{4},{1,2}}
{{1},{1,2,3}}
{{1,2},{1,3}}
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@@#, And@@UnsameQ@@@#, Length[#]==k]&]], {n, 0, 10}, {k, 0, n}]
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(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
Row sums are A050342.
Column k = 1 is A000009.
Sequence in context: A303708 A319138 A349396 * A281081 A103344 A123484
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Dec 18 2019
STATUS
approved

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Last modified March 19 01:57 EDT 2024. Contains 370952 sequences. (Running on oeis4.)