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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.

%I A107329 #20 Oct 28 2021 18:41:22
%S A107329 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,
%T A107329 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,
%U A107329 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
%N 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.
%C A107329 T(n,n) equals A066882(n).
%H A107329 Alois P. Heinz, <a href="/A107329/b107329.txt">rows n = 1..200, flattened</a>
%F A107329 T(n,k) is coefficient of x^k in 1/Product(1-x^p_i) with p_i the prime factors of n.
%e A107329 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].
%e A107329 Triangle begins:
%e A107329   {0},
%e A107329   {0, 1},
%e A107329   {0, 0, 1},
%e A107329   {0, 1, 0, 1},
%e A107329   {0, 0, 0, 0, 1},
%e A107329   {0, 1, 1, 1, 1, 2},
%e A107329   {0, 0, 0, 0, 0, 0, 1},
%e A107329   {0, 1, 0, 1, 0, 1, 0, 1},
%e A107329   {0, 0, 1, 0, 0, 1, 0, 0, 1},
%e A107329   ...
%p A107329 with(numtheory):
%p A107329 T:= proc(n) local b, l; l:= sort([factorset(n)[]]):
%p A107329       b:= proc(m, i) option remember; `if`(m=0, 1, `if`(i<1, 0,
%p A107329             b(m, i-1)+`if`(l[i]>m, 0, b(m-l[i], i))))
%p A107329           end; forget(b):
%p A107329       seq(b(k, nops(l)), k=1..n)
%p A107329     end:
%p A107329 seq(T(n), n=1..20);  # _Alois P. Heinz_, Oct 28 2021
%t A107329 Table[Rest@CoefficientList[Series[1/Times @@ ((1-x^#)& /@ (First /@ FactorInteger[n])), {x, 0, n}], x], {n, 2, 24}]
%Y A107329 Cf. A066882.
%Y A107329 Row sums +1 give A092976.
%K A107329 easy,nonn,look,tabl
%O A107329 1,21
%A A107329 _Wouter Meeussen_, May 22 2005
%E A107329 T(1,1) = 0 prepended by _Michel Marcus_, Oct 28 2021