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%I #19 Dec 31 2018 07:31:57
%S 1,1,1,1,1,2,1,1,2,2,1,1,2,4,3,1,1,2,4,6,2,1,1,2,4,8,11,4,1,1,2,4,8,
%T 14,17,2,1,1,2,4,8,16,27,29,4,1,1,2,4,8,16,30,49,47,3,1,1,2,4,8,16,32,
%U 59,92,78,4,1,1,2,4,8,16,32,62,113,170,130,2
%N Number A(n,k) of compositions of n where differences between neighboring parts are in {-k,...,k}; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%H Alois P. Heinz, <a href="/A214248/b214248.txt">Antidiagonals n = 0..140, flattened</a>
%e A(3,0) = 2: [3], [1,1,1].
%e A(4,1) = 6: [4], [2,2], [2,1,1], [1,2,1], [1,1,2], [1,1,1,1].
%e A(5,2) = 14: [5], [3,2], [3,1,1], [2,3], [2,2,1], [2,1,2], [2,1,1,1], [1,3,1], [1,2,2], [1,2,1,1], [1,1,3], [1,1,2,1], [1,1,1,2], [1,1,1,1,1].
%e Square array A(n,k) begins:
%e 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 2, 2, 2, 2, 2, 2, 2, 2, ...
%e 2, 4, 4, 4, 4, 4, 4, 4, ...
%e 3, 6, 8, 8, 8, 8, 8, 8, ...
%e 2, 11, 14, 16, 16, 16, 16, 16, ...
%e 4, 17, 27, 30, 32, 32, 32, 32, ...
%e 2, 29, 49, 59, 62, 64, 64, 64, ...
%p b:= proc(n, i, k) option remember;
%p `if`(n<1 or i<1, 0, `if`(n=i, 1, add(b(n-i, i+j, k), j={$-k..k})))
%p end:
%p A:= (n, k)-> `if`(n=0, 1, add(b(n, j, k), j=1..n)):
%p seq(seq(A(n, d-n), n=0..d), d=0..15);
%t 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, -k, 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 Columns k=0-2 give: A000005, A034297, A214255.
%Y Main diagonal gives: A011782.
%Y Cf. A214246, A214247, A214249, A214257, A214258, A214268, A214269.
%K nonn,tabl
%O 0,6
%A _Alois P. Heinz_, Jul 08 2012
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