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%I #14 Feb 25 2017 03:00:22
%S 1,1,2,20,48,264,4296,14528,89472,593248,19115360,75604544,599169408,
%T 4141674240,40147321344,2159264715776,10240251475456,92926573965184,
%U 746025520714112,7285397378650112,82900557619046912,7796186873306241024,41825012467664893440
%N Number of compositions of n into distinct parts where each part i is marked with a word of length i over a k-ary alphabet (k=1,2,3,...) whose letters appear in alphabetical order and all k letters occur at least once in the composition.
%C Also number of matrices with nonnegative integer entries and without zero rows or columns such that sum of all entries is equal to n and the column sums are distinct.
%C a(2) = 2:
%C [1] [2]
%C [1]
%H Alois P. Heinz, <a href="/A261838/b261838.txt">Table of n, a(n) for n = 0..300</a>
%e a(0) = 1: the empty composition.
%e a(1) = 1: 1a.
%e a(2) = 2: 2aa (for k=1), 2ab (for k=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-> add(add(b(n$2, 0, k-i)*(-1)^i*binomial(k, i), i=0..k), k=0..n):
%p seq(a(n), n=0..25);
%t b[n_, i_, p_, k_] := b[n, i, p, k] = If[i*(i+1)/2 < n, 0, If[n == 0, p!, b[n, i-1, p, k] + If[i>n, 0, b[n-i, i-1, p+1, k]*Binomial[i+k-1, k-1]]]]; a[n_] := Sum[b[n, n, 0, k-i]*(-1)^i*Binomial[k, i], {k, 0, n}, {i, 0, k}]; Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Feb 25 2017, translated from Maple *)
%Y Row sums of A261836.
%Y Cf. A120733.
%K nonn
%O 0,3
%A _Alois P. Heinz_, Sep 02 2015
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