%I #6 Sep 03 2015 11:49:39

%S 1,10,15,40,183,266,549,1056,4421,5850,12245,20644,39809,141818,

%T 195421,370808,633379,1126518,1870135,6531964,8547045,16324018,

%U 26458275,46612364,73200021,127916094,385244951,518151276,939317459,1516648678,2564211485,4008404972

%N Number of compositions of n into distinct parts where each part i is marked with a word of length i over a binary alphabet whose letters appear in alphabetical order and all letters occur at least once in the composition.

%C Also number of matrices with two 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="/A261853/b261853.txt">Table of n, a(n) for n = 2..2500</a>

%F a(n) = A261836(n,2).

%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))(2):

%p seq(a(n), n=2..40);

%Y Column k=2 of A261836.

%K nonn

%O 2,2

%A _Alois P. Heinz_, Sep 03 2015