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A214246 Number A(n,k) of compositions of n where differences between neighboring parts are in {-k,0,k}; square array A(n,k), n>=0, k>=0, read by antidiagonals.

%I A214246 #20 Dec 31 2018 07:30:39
%S A214246 1,1,1,1,1,2,1,1,2,2,1,1,2,4,3,1,1,2,2,6,2,1,1,2,2,5,11,4,1,1,2,2,3,5,
%T A214246 17,2,1,1,2,2,3,4,10,29,4,1,1,2,2,3,2,7,10,47,3,1,1,2,2,3,2,6,8,21,78,
%U A214246 4,1,1,2,2,3,2,4,5,9,22,130,2
%N A214246 Number A(n,k) of compositions of n where differences between neighboring parts are in {-k,0,k}; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%H A214246 Alois P. Heinz, <a href="/A214246/b214246.txt">Antidiagonals n = 0..140, flattened</a>
%e A214246 A(3,0) = 2: [3], [1,1,1].
%e A214246 A(4,1) = 6: [4], [2,2], [2,1,1], [1,2,1], [1,1,2], [1,1,1,1].
%e A214246 A(5,2) = 5: [5], [3,1,1], [1,3,1], [1,1,3], [1,1,1,1,1].
%e A214246 A(6,3) = 7: [6], [4,1,1], [3,3], [2,2,2], [1,4,1], [1,1,4], [1,1,1,1,1,1].
%e A214246 Square array A(n,k) begins:
%e A214246   1,  1,  1,  1,  1,  1,  1,  1, ...
%e A214246   1,  1,  1,  1,  1,  1,  1,  1, ...
%e A214246   2,  2,  2,  2,  2,  2,  2,  2, ...
%e A214246   2,  4,  2,  2,  2,  2,  2,  2, ...
%e A214246   3,  6,  5,  3,  3,  3,  3,  3, ...
%e A214246   2, 11,  5,  4,  2,  2,  2,  2, ...
%e A214246   4, 17, 10,  7,  6,  4,  4,  4, ...
%e A214246   2, 29, 10,  8,  5,  4,  2,  2, ...
%p A214246 b:= proc(n, i, k) option remember;
%p A214246       `if`(n<1 or i<1, 0, `if`(n=i, 1, add(b(n-i, i+j, k), j={-k, 0, k})))
%p A214246     end:
%p A214246 A:= (n, k)-> `if`(n=0, 1, add(b(n, j, k), j=1..n)):
%p A214246 seq(seq(A(n, d-n), n=0..d), d=0..15);
%t A214246 b[n_, i_, k_] := b[n, i, k] = If[n < 1 || i < 1, 0, If[n == i, 1, Sum[b[n - i, i + j, k], {j, Union[{-k, 0, k}]}]]]; A[n_, k_] := If[n == 0, 1, Sum[b[n, j, k], {j, 1, n}]]; Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 15}] // Flatten (* _Jean-François Alcover_, Dec 27 2013, translated from Maple *)
%Y A214246 Column k=0 and main diagonal give: A000005.
%Y A214246 Columns k=1, 2 give: A034297, A214253.
%Y A214246 Cf. A214247, A214248, A214249, A214257, A214258, A214268, A214269.
%K A214246 nonn,tabl
%O A214246 0,6
%A A214246 _Alois P. Heinz_, Jul 08 2012