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%I #4 Sep 03 2015 12:14:03
%S 9,92,1562,3908,14791,50208,540552,987120,3138143,7862580,23436690,
%T 204455140,349297653,956040232,2228084512,5599922904,13449425997,
%U 116772809532,182990434794,483410072060,1033025269277,2455590595520,5184309618676,12755194552152
%N Number of compositions of n into distinct parts where each part i is marked with a word of length i over a quaternary alphabet whose letters appear in alphabetical order and all letters occur at least once in the composition.
%C Also number of matrices with four 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="/A261855/b261855.txt">Table of n, a(n) for n = 4..2500</a>
%F a(n) = A261836(n,4).
%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))(4):
%p seq(a(n), n=4..40);
%Y Column k=4 of A261836.
%K nonn
%O 4,1
%A _Alois P. Heinz_, Sep 03 2015
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