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A261960 Number A(n,k) of compositions of n such that no part equals any of its k immediate predecessors; square array A(n,k), n>=0, k>=0, read by antidiagonals.

%I A261960 #16 Oct 28 2018 11:04:41
%S A261960 1,1,1,1,1,2,1,1,1,4,1,1,1,3,8,1,1,1,3,4,16,1,1,1,3,3,7,32,1,1,1,3,3,
%T A261960 5,14,64,1,1,1,3,3,5,11,23,128,1,1,1,3,3,5,11,15,39,256,1,1,1,3,3,5,
%U A261960 11,13,23,71,512,1,1,1,3,3,5,11,13,19,37,124,1024
%N A261960 Number A(n,k) of compositions of n such that no part equals any of its k immediate predecessors; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%H A261960 Alois P. Heinz, <a href="/A261960/b261960.txt">Antidiagonals n = 0..50, flattened</a>
%e A261960 Square array A(n,k) begins:
%e A261960 :  1,  1,  1,  1,  1,  1,  1, ...
%e A261960 :  1,  1,  1,  1,  1,  1,  1, ...
%e A261960 :  2,  1,  1,  1,  1,  1,  1, ...
%e A261960 :  4,  3,  3,  3,  3,  3,  3, ...
%e A261960 :  8,  4,  3,  3,  3,  3,  3, ...
%e A261960 : 16,  7,  5,  5,  5,  5,  5, ...
%e A261960 : 32, 14, 11, 11, 11, 11, 11, ...
%p A261960 b:= proc(n, l) option remember;
%p A261960       `if`(n=0, 1, add(`if`(j in l, 0, b(n-j,
%p A261960       `if`(l=[], [], [subsop(1=NULL, l)[], j]))), j=1..n))
%p A261960     end:
%p A261960 A:= (n, k)-> b(n, [0$min(n, k)]):
%p A261960 seq(seq(A(n, d-n), n=0..d), d=0..12);
%t A261960 b[n_, l_] := b[n, l] = If[n==0, 1, Sum[If[MemberQ[l, j], 0, b[n-j, If[l == {}, {}, Append[Rest[l], j]]]], {j, 1, n}]]; A[n_, k_] := b[n, Array[0&, Min[n, k]]]; Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* _Jean-François Alcover_, Feb 08 2017, translated from Maple *)
%Y A261960 Columns k=0-2 give: A011782, A003242, A261962.
%Y A261960 Main diagonal gives A032020.
%Y A261960 Cf. A261959, A261981.
%K A261960 nonn,tabl
%O A261960 0,6
%A A261960 _Alois P. Heinz_, Sep 06 2015