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A214249 Number A(n,k) of compositions of n where differences between neighboring parts are in {-k,...,k} \ {0}; square array A(n,k), n>=0, k>=0, read by antidiagonals.

%I A214249 #21 Dec 31 2018 07:32:10
%S A214249 1,1,1,1,1,1,1,1,1,1,1,1,1,3,1,1,1,1,3,2,1,1,1,1,3,4,4,1,1,1,1,3,4,5,
%T A214249 5,1,1,1,1,3,4,7,11,5,1,1,1,1,3,4,7,12,14,7,1,1,1,1,3,4,7,14,20,18,10,
%U A214249 1,1,1,1,3,4,7,14,21,30,36,9,1,1,1,1,3,4,7,14,23,36,50,49,14,1
%N A214249 Number A(n,k) of compositions of n where differences between neighboring parts are in {-k,...,k} \ {0}; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%H A214249 Alois P. Heinz, <a href="/A214249/b214249.txt">Rows n = 0..140, flattened</a>
%e A214249 A(3,0) = 1: [3].
%e A214249 A(4,1) = 2: [4], [1,2,1].
%e A214249 A(5,2) = 5: [5], [3,2], [2,3], [2,1,2], [1,3,1].
%e A214249 A(6,3) = 12: [6], [4,2], [3,2,1], [3,1,2], [2,4], [2,3,1], [2,1,3], [2,1,2,1], [1,4,1], [1,3,2], [1,2,3], [1,2,1,2].
%e A214249 Square array A(n,k) begins:
%e A214249   1,  1,  1,  1,  1,  1,  1,  1, ...
%e A214249   1,  1,  1,  1,  1,  1,  1,  1, ...
%e A214249   1,  1,  1,  1,  1,  1,  1,  1, ...
%e A214249   1,  3,  3,  3,  3,  3,  3,  3, ...
%e A214249   1,  2,  4,  4,  4,  4,  4,  4, ...
%e A214249   1,  4,  5,  7,  7,  7,  7,  7, ...
%e A214249   1,  5, 11, 12, 14, 14, 14, 14, ...
%e A214249   1,  5, 14, 20, 21, 23, 23, 23, ...
%p A214249 b:= proc(n, i, k) option remember; `if`(n<1 or i<1, 0,
%p A214249       `if`(n=i, 1, add(b(n-i, i+j, k), j={$-k..k} minus{0})))
%p A214249     end:
%p A214249 A:= (n, k)-> `if`(n=0, 1, add(b(n, j, min(n, k)), j=1..n)):
%p A214249 seq(seq(A(n, d-n), n=0..d), d=0..15);
%t A214249 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, Range[-k, -1] ~Join~ Range[k]}]]]; A[n_, k_] := If[n == 0, 1, Sum[b[n, j, Min[n, k]], {j, 1, n}]]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 15}] // Flatten (* _Jean-François Alcover_, Jan 15 2014, translated from Maple *)
%Y A214249 Columns k=0-2 give: A000012, A173258, A214256.
%Y A214249 Main diagonal gives: A003242.
%Y A214249 Cf. A214246, A214247, A214248, A214257, A214258, A214268, A214269.
%K A214249 nonn,tabl
%O A214249 0,14
%A A214249 _Alois P. Heinz_, Jul 08 2012