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%I #16 Apr 29 2018 02:13:21
%S 1,0,1,6,1,72,6,2320,72,245765,2320,151182379,245765
%N Triangle read by rows: T(n,k) = number of increasing sequences of n positive integers with reciprocals adding up to k (k=1,2,...,A055980(n)).
%C T(n,k) = 0 for all k > A055980(n).
%C For n=3,...,11, we have T(n,2) = T(n-1,1). However, T(12,2) > T(11,1).
%C Conjecture: for n in A115515 (i.e., A055980(n+1)=A055980(n)+1), the sequences being enumerated by T(n,A055980(n)) must start with 1. E.g., there is no 10-tuple (x_1,x_2,...,x_10) with 1 < x_1 < ... < x_10 and 1/x_1 + ... + 1/x_10 = 2 (=A055980(10)).
%e Triangle starts with:
%e n=1: 1
%e n=2: 0
%e n=3: 1
%e n=4: 6, 1
%e n=5: 72, 6
%e n=6: 2320, 72
%e n=7: 245765, 2320
%e n=8: 151182379, 245765
%e ...
%Y Cf. A280518 (row sums), A006585 (column k=1), A156869 (nondecreasing sequences), A280519 (ordered sequences).
%K nonn,tabf,more
%O 1,4
%A _Max Alekseyev_, Jan 04 2017
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