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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.
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%I #4 Sep 03 2015 04:29:56

%S 1,3,6,46,75,231,1414,2376,5985,14151,89454,135330,343677,697017,

%T 1657212,9439826,14381055,33119667,66361286,141451860,283907499,

%U 1642516411,2346737106,5367877296,10093521943,20923900623,38428831710,80538197724,416229711735

%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.

%C Also matrices with three rows of nonnegative integers with distinct positive column sums and total element sum n.

%H Alois P. Heinz, <a href="/A261841/b261841.txt">Table of n, a(n) for n = 0..5000</a>

%p b:= proc(n, i, p) option remember;

%p `if`(i*(i+1)/2<n, 0, `if`(n=0, p!, b(n, i-1, p)+

%p `if`(i>n, 0, b(n-i, i-1, p+1)*binomial(i+2, 2))))

%p end:

%p a:= n-> b(n$2, 0):

%p seq(a(n), n=0..40);

%Y Column k=3 of A261835.

%K nonn

%O 0,2

%A _Alois P. Heinz_, Sep 03 2015