A249771 - OEIS

%I #25 Dec 29 2018 13:02:10

%S 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 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 3,5,1,2

%N 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 The length of n-th row is A051282(n).

%C Signatures differing only by a (trailing) list of ones give identical rows.

%H Álvar Ibeas, <a href="/A249771/b249771.txt">Rows n=2..1075, flattened</a>

%F 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 T(n,k) = A249770(A025487(n),k).

%F T(n,1) + T(n,2) = A052304(n).

%e First rows:

%e 1;

%e 1,1;

%e 1;

%e 1,1,1;

%e 1,1;

%e 1,2,1,1;

%e 1,1,1;

%e 1;

%e 1,2,2,1,1;

%e 1,3;

%e ...

%Y Refinement of A050360. Last row elements: A249773. Cf. A249770, A052304.

%K nonn,tabf

%O 2,11

%A _Álvar Ibeas_, Nov 06 2014