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%I #35 Mar 24 2020 22:30:25
%S 1,0,1,0,0,1,0,0,1,1,0,0,0,0,1,0,0,1,3,2,1,0,0,0,0,0,0,1,0,0,0,1,3,3,
%T 2,1,0,0,0,0,2,2,3,2,1,0,0,0,1,3,6,7,5,3,1,0,0,0,0,0,0,0,0,0,0,1,0,0,
%U 0,3,8,15,21,24,20,11,4,1
%N Triangle read by rows: T(n,k) is the number of Egyptian fractions 1 = 1/x_1 + ... + 1/x_k , with 0 < x_1 <= ... <= x_k = n.
%H <a href="/index/Ed#Egypt">Index entries for sequences related to Egyptian fractions</a>
%F T(n,n) = 1.
%F If n is prime, T(n,k) = 0 for 1 <= k < n.
%e 1 = 1/2 + 1/6 + 1/6 + 1/6 = 1/3 + 1/3 + 1/6 + 1/6 = 1/3 + 1/4 + 1/4 + 1/6. So T(6,4) = 3.
%e Triangle begins:
%e n\k | 1 2 3 4 5 6 7 8 9 10 11 12
%e -----+----------------------------------------
%e 1 | 1;
%e 2 | 0, 1;
%e 3 | 0, 0, 1;
%e 4 | 0, 0, 1, 1;
%e 5 | 0, 0, 0, 0, 1;
%e 6 | 0, 0, 1, 3, 2, 1;
%e 7 | 0, 0, 0, 0, 0, 0, 1;
%e 8 | 0, 0, 0, 1, 3, 3, 2, 1;
%e 9 | 0, 0, 0, 0, 2, 2, 3, 2, 1;
%e 10 | 0, 0, 0, 1, 3, 6, 7, 5, 3, 1;
%e 11 | 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
%e 12 | 0, 0, 0, 3, 8, 15, 21, 24, 20, 11, 4, 1;
%Y Row sums give A092666.
%Y Cf. A020473, A333496.
%K nonn,tabl
%O 1,19
%A _Seiichi Manyama_, Mar 24 2020