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%I #13 Apr 14 2022 07:23:10
%S 1,1,1,1,2,1,1,2,3,1,1,3,4,4,1,1,3,6,7,5,1,1,4,7,11,11,6,1,1,4,9,15,
%T 19,16,7,1,1,5,11,19,29,31,22,8,1,1,5,13,25,39,52,48,29,9,1,1,6,15,30,
%U 53,76,88,71,37,10,1,1,6,18,37,67,107,140,142,101,46,11,1,1,7,20,44,84,143,207,245,220,139,56,12,1
%N Triangle, T(n,k) = number of compositions a(1),...,a(k) of n, such that a(i+1) <= a(i) + 1 for 1 <= i < k.
%H Alois P. Heinz, <a href="/A168443/b168443.txt">Rows n = 1..200, flattened</a>
%e Triangle T(n,k) begins:
%e 1;
%e 1, 1;
%e 1, 2, 1;
%e 1, 2, 3, 1;
%e 1, 3, 4, 4, 1;
%e 1, 3, 6, 7, 5, 1;
%e 1, 4, 7, 11, 11, 6, 1;
%e 1, 4, 9, 15, 19, 16, 7, 1;
%e ...
%p b:= proc(n, k) option remember; expand(`if`(n=0, 1,
%p x*add(b(n-j, j), j=1..min(n, k+1))))
%p end:
%p T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n$2)):
%p seq(T(n), n=1..14); # _Alois P. Heinz_, Jan 21 2022
%t b[n_, k_] := b[n, k] = Expand[If[n == 0, 1,
%t x*Sum[b[n - j, j], {j, 1, Min[n, k + 1]}]]];
%t T[n_] := Rest@CoefficientList[b[n, n], x];
%t Table[T[n], {n, 1, 14}] // Flatten (* _Jean-François Alcover_, Apr 14 2022, after _Alois P. Heinz_ *)
%Y Cf. A003116 (row sums), A168396.
%K nonn,tabl
%O 1,5
%A _Vladeta Jovovic_, Nov 25 2009
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