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A330459
Number of set partitions of set-systems with total sum n.
10
1, 1, 1, 4, 6, 11, 26, 42, 78, 148, 280, 481, 867, 1569, 2742, 4933, 8493, 14857, 25925, 44877, 77022, 132511, 226449, 385396, 657314, 1111115, 1875708, 3157379, 5309439, 8885889, 14861478, 24760339, 41162971, 68328959, 113099231, 186926116, 308230044
OFFSET
0,4
COMMENTS
Number of sets of disjoint nonempty sets of nonempty sets of positive integers with total sum n.
LINKS
FORMULA
a(n) = Sum_k A330462(n,k) * A000110(k).
EXAMPLE
The a(6) = 26 partitions:
((6)) ((15)) ((123)) ((1)(2)(12))
((24)) ((1)(14)) ((1))((2)(12))
((1)(5)) ((1)(23)) ((12))((1)(2))
((2)(4)) ((2)(13)) ((2))((1)(12))
((1))((5)) ((3)(12)) ((1))((2))((12))
((2))((4)) ((1))((14))
((1))((23))
((1)(2)(3))
((2))((13))
((3))((12))
((1))((2)(3))
((2))((1)(3))
((3))((1)(2))
((1))((2))((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, 3], And[UnsameQ@@Join@@#, And@@UnsameQ@@@Join@@#]&]], {n, 0, 10}]
PROG
(PARI) \\ here L is A000009 and BellP is A000110 as series.
L(n)={eta(x^2 + O(x*x^n))/eta(x + O(x*x^n))}
BellP(n)={serlaplace(exp( exp(x + O(x*x^n)) - 1))}
seq(n)={my(c=L(n), b=BellP(n), v=Vec(prod(k=1, n, (1 + x^k*y + O(x*x^n))^polcoef(c, k)))); vector(#v, n, my(r=v[n]); sum(k=0, n-1, polcoeff(b, k)*polcoef(r, k)))} \\ Andrew Howroyd, Dec 29 2019
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 17 2019
EXTENSIONS
Terms a(18) and beyond from Andrew Howroyd, Dec 29 2019
STATUS
approved