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A107329
Triangle read by rows: T(n,k) gives number of partitions of k, (k=1..n) into the prime factors of n, for n>=1.
2
0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 2, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2
OFFSET
1,21
COMMENTS
T(n,n) equals A066882(n).
LINKS
FORMULA
T(n,k) is coefficient of x^k in 1/Product(1-x^p_i) with p_i the prime factors of n.
EXAMPLE
T(30,12)=5 counting [2,2,2,2,2,2], [2,2,2,3,3], [3,3,3,3], [2,2,3,5] and [2,5,5].
Triangle begins:
{0},
{0, 1},
{0, 0, 1},
{0, 1, 0, 1},
{0, 0, 0, 0, 1},
{0, 1, 1, 1, 1, 2},
{0, 0, 0, 0, 0, 0, 1},
{0, 1, 0, 1, 0, 1, 0, 1},
{0, 0, 1, 0, 0, 1, 0, 0, 1},
...
MAPLE
with(numtheory):
T:= proc(n) local b, l; l:= sort([factorset(n)[]]):
b:= proc(m, i) option remember; `if`(m=0, 1, `if`(i<1, 0,
b(m, i-1)+`if`(l[i]>m, 0, b(m-l[i], i))))
end; forget(b):
seq(b(k, nops(l)), k=1..n)
end:
seq(T(n), n=1..20); # Alois P. Heinz, Oct 28 2021
MATHEMATICA
Table[Rest@CoefficientList[Series[1/Times @@ ((1-x^#)& /@ (First /@ FactorInteger[n])), {x, 0, n}], x], {n, 2, 24}]
CROSSREFS
Cf. A066882.
Row sums +1 give A092976.
Sequence in context: A096563 A216512 A078359 * A263717 A230279 A376366
KEYWORD
easy,nonn,look,tabl
AUTHOR
Wouter Meeussen, May 22 2005
EXTENSIONS
T(1,1) = 0 prepended by Michel Marcus, Oct 28 2021
STATUS
approved