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%I #38 Dec 29 2018 13:02:17
%S 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,
%T 1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,1,1,1,1,1,1,3,1,1,1,1,1,1,1,1,1,1,1,1,
%U 1,1,1
%N Irregular triangle read by rows: T(n,k) is the number of Abelian groups of order n with k invariant factors (2 <= n, 1 <= k).
%C The length of n-th row is A051903(n) and its last element is A249773(A101296(n)).
%C T(n,k) depends only on k and the prime signature of n.
%H Álvar Ibeas, <a href="/A249770/b249770.txt">Rows n=2..58654, flattened</a>
%F T(n,k) = A249771(A101296(n),k).
%F T(n,1) = 1. If k > 1 and n = Product(p_i^e_i), T(n,k) = Sum(Product(A008284(e_i,k), i in I) * Product(A026820(e_i,k-1), i not in I)), where the sum is taken over nonempty subsets I of {1,...,omega(n)}.
%F If p is prime and gcd(p,n) = 1, T(pn,k) = T(n,k).
%F Dirichlet g.f. of column sums: zeta(s)zeta(2s)···zeta(ms) = 1 + Sum_{n >= 2} (Sum_{k=1..m} T(n,k)) / n^s.
%F T(n,1) + T(n,2) = A046951(n)
%e First rows:
%e 1;
%e 1;
%e 1,1;
%e 1;
%e 1;
%e 1;
%e 1,1,1;
%e 1,1;
%e 1;
%e 1;
%e 1,1;
%e 1;
%e 1;
%e 1;
%e 1,2,1,1;
%e 1;
%e ...
%t f[{x_, y_}] := x^IntegerPartitions[y];
%t g[n_] := FactorInteger[n][[1, 1]];
%t h[list_] := Apply[Times,Map[PadRight[#, Max[Map[Length, SplitBy[list, g]]], 1] &,SplitBy[list, g]]]; t[list_] := Tally[Map[Length, list]][[All, 2]];
%t Map[t, Table[Map[h, Join @@@ Tuples[Map[f, FactorInteger[n]]]], {n, 2, 50}]] // Grid (* _Geoffrey Critzer_, Nov 26 2015 *)
%Y Refinement of A000688.
%Y Cf. A008284, A051903, A026820, A101296, A249771, A249773, A046951, A264809.
%K nonn,tabf
%O 2,21
%A _Álvar Ibeas_, Nov 06 2014
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