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A322480 Irregular triangular array read by rows: T(n,k), n>=1, is the number of ordered factorizations corresponding to each unordered factorization, indexed by k.

%I A322480 #38 Feb 14 2019 12:39:04
%S A322480 1,1,1,1,1,1,1,2,1,1,2,1,1,1,1,2,1,1,2,2,3,1,1,2,1,2,1,2,1,3,1,1,1,2,
%T A322480 2,3,1,1,2,2,3,1,2,1,2,1,1,2,2,2,3,6,4,1,1,1,2,1,2,1,1,2,2,3,1,1,2,2,
%U A322480 2,6,1,1,2,2,3,3,4,1,1,2,1,2,1,2,1,2,2,2,3,1,6,3,6,1,1,2,1,2,1,2,2,3,2,6,4,1,1,2,2,2,6,1,1,2,2,3,1,2,2,3,1,2,1,1,2,2,2,3,2,6,6,4,12,5
%N A322480 Irregular triangular array read by rows: T(n,k), n>=1, is the number of ordered factorizations corresponding to each unordered factorization, indexed by k.
%C A322480 The method of indexing the unordered factorizations of n in this array is as follows: take all unordered factorizations of n and write them with their factors in nonincreasing order (e.g., 2*4*5*3 becomes 5*4*3*2), and order these reverse-lexicographically (e.g., for 12: 12, 6*2, 4*3, 3*2*2), then assign the index k to the k-th factorization in this ordering.
%C A322480 For a sequence f with Dirichlet inverse f^(-1), f^(-1)(n) is the sum over all multisets M of integers > 1 with product n, of the product of the terms f(m) with indices m in M (counted with multiplicity) multiplied by T(n,k)*(-1)^c/f(1)^(c+1) where c = |M| and T(n,k) corresponds to M.
%C A322480 The multiset of entries in the n-th row is determined by the prime signature of n.
%C A322480 For the p^j-th row with p a prime, the entries give the number of compositions of j corresponding to each partition of j, indexed by k in an analogous manner, given by the j-th row of A048996.
%e A322480   1;
%e A322480   1;
%e A322480   1;
%e A322480   1, 1;
%e A322480   1;
%e A322480   1, 2;
%e A322480   1;
%e A322480   1, 2, 1;
%e A322480   1, 1;
%e A322480   1, 2;
%e A322480   1;
%e A322480   1, 2, 2, 3;
%e A322480   etc.
%e A322480 The 12th row is 1,2,2,3, because 12 can be factored as 12, 6*2, 3*4 or 3*2*2 with respective sets of ordered factorizations {12}, {6*2, 2*6}, {4*3, 3*4} and {3*2*2, 2*3*2, 2*2*3}, with respective cardinalities 1, 2, 2 and 3.
%Y A322480 Cf. A048996, A002033 (row sums), A212171, A251683, A001055 (row lengths).
%K A322480 nonn,tabf
%O A322480 1,8
%A A322480 _Thomas Anton_, Dec 09 2018