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A249771 Irregular triangle read by rows: T(n,k) is the number of Abelian groups of order A025487(n) with k invariant factors (2 <= n, 1 <= k).

%I A249771 #25 Dec 29 2018 13:02:10
%S A249771 1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,2,2,1,1,1,3,1,2,1,1,1,1,1,3,3,2,
%T A249771 1,1,1,3,2,1,2,2,1,1,1,1,1,1,3,4,3,2,1,1,1,5,2,2,1,3,1,3,3,2,1,1,1,1,
%U A249771 3,5,1,2
%N A249771 Irregular triangle read by rows: T(n,k) is the number of Abelian groups of order A025487(n) with k invariant factors (2 <= n, 1 <= k).
%C A249771 The length of n-th row is A051282(n).
%C A249771 Signatures differing only by a (trailing) list of ones give identical rows.
%H A249771 Álvar Ibeas, <a href="/A249771/b249771.txt">Rows n=2..1075, flattened</a>
%F A249771 T(n,1) = 1. If k > 1 and the prime signature is (e_1,...,e_s), 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,...,s}.
%F A249771 T(n,k) = A249770(A025487(n),k).
%F A249771 T(n,1) + T(n,2) = A052304(n).
%e A249771 First rows:
%e A249771 1;
%e A249771 1,1;
%e A249771 1;
%e A249771 1,1,1;
%e A249771 1,1;
%e A249771 1,2,1,1;
%e A249771 1,1,1;
%e A249771 1;
%e A249771 1,2,2,1,1;
%e A249771 1,3;
%e A249771 ...
%Y A249771 Refinement of A050360. Last row elements: A249773. Cf. A249770, A052304.
%K A249771 nonn,tabf
%O A249771 2,11
%A A249771 _Álvar Ibeas_, Nov 06 2014