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A168443 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.

%I A168443 #13 Apr 14 2022 07:23:10
%S A168443 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 A168443 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 A168443 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 A168443 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 A168443 Alois P. Heinz, <a href="/A168443/b168443.txt">Rows n = 1..200, flattened</a>
%e A168443 Triangle T(n,k) begins:
%e A168443   1;
%e A168443   1, 1;
%e A168443   1, 2, 1;
%e A168443   1, 2, 3,  1;
%e A168443   1, 3, 4,  4,  1;
%e A168443   1, 3, 6,  7,  5,  1;
%e A168443   1, 4, 7, 11, 11,  6, 1;
%e A168443   1, 4, 9, 15, 19, 16, 7, 1;
%e A168443   ...
%p A168443 b:= proc(n, k) option remember; expand(`if`(n=0, 1,
%p A168443       x*add(b(n-j, j), j=1..min(n, k+1))))
%p A168443     end:
%p A168443 T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n$2)):
%p A168443 seq(T(n), n=1..14);  # _Alois P. Heinz_, Jan 21 2022
%t A168443 b[n_, k_] := b[n, k] = Expand[If[n == 0, 1,
%t A168443      x*Sum[b[n - j, j], {j, 1, Min[n, k + 1]}]]];
%t A168443 T[n_] := Rest@CoefficientList[b[n, n], x];
%t A168443 Table[T[n], {n, 1, 14}] // Flatten (* _Jean-François Alcover_, Apr 14 2022, after _Alois P. Heinz_ *)
%Y A168443 Cf. A003116 (row sums), A168396.
%K A168443 nonn,tabl
%O A168443 1,5
%A A168443 _Vladeta Jovovic_, Nov 25 2009