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%I #4 Sep 03 2015 12:03:36
%S 7,21,96,832,1539,4281,10902,76020,117585,306639,634686,1537206,
%T 9013319,13793487,32005392,64458596,138068775,278292429,1622912266,
%U 2321086080,5318890971,10014128239,20784037248,38209197732,80154402633,415073903937,593664848658
%N Number of compositions of n into distinct parts where each part i is marked with a word of length i over a ternary alphabet whose letters appear in alphabetical order and all letters occur at least once in the composition.
%C Also number of matrices with three rows of nonnegative integer entries and without zero rows or columns such that the sum of all entries is equal to n and the column sums are distinct.
%H Alois P. Heinz, <a href="/A261854/b261854.txt">Table of n, a(n) for n = 3..2500</a>
%F a(n) = A261836(n,3):
%p b:= proc(n, i, p, k) option remember;
%p `if`(i*(i+1)/2<n, 0, `if`(n=0, p!, b(n, i-1, p, k)+
%p `if`(i>n, 0, b(n-i, i-1, p+1, k)*binomial(i+k-1, k-1))))
%p end:
%p a:= n->(k->add(b(n$2, 0, k-i)*(-1)^i*binomial(k, i), i=0..k))(3):
%p seq(a(n), n=3..40);
%Y Column k=3 of A261836.
%K nonn
%O 3,1
%A _Alois P. Heinz_, Sep 03 2015
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