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A214247 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. 11

%I

%S 1,1,1,1,1,2,1,1,1,2,1,1,1,3,3,1,1,1,1,2,2,1,1,1,1,3,4,4,1,1,1,1,1,2,

%T 5,2,1,1,1,1,1,3,3,5,4,1,1,1,1,1,1,2,2,7,3,1,1,1,1,1,1,3,3,6,10,4,1,1,

%U 1,1,1,1,1,2,1,4,9,2

%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="/A214247/b214247.txt">Antidiagonals n = 0..140</a>

%e A(5,0) = 2: [5], [1,1,1,1,1].

%e A(5,1) = 4: [5], [3,2], [2,3], [2,1,2].

%e A(5,2) = 2: [5], [1,3,1].

%e A(5,3) = 3: [5], [4,1], [1,4].

%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, 1, 1, 1, 1, 1, 1, 1, ...

%e 2, 3, 1, 1, 1, 1, 1, 1, ...

%e 3, 2, 3, 1, 1, 1, 1, 1, ...

%e 2, 4, 2, 3, 1, 1, 1, 1, ...

%e 4, 5, 3, 2, 3, 1, 1, 1, ...

%e 2, 5, 2, 3, 2, 3, 1, 1, ...

%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, Union[{-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-Fran├žois Alcover_, Dec 13 2013, translated from Maple *)

%Y Columns k=0-2 give: A000005, A173258, A214254.

%Y Rows n=0, 1 and main diagonal give: A000012.

%Y Cf. A214246, A214248, A214249, A214257, A214258, A214268, A214269.

%K nonn,tabl,look

%O 0,6

%A _Alois P. Heinz_, Jul 08 2012

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Last modified July 9 22:15 EDT 2020. Contains 335545 sequences. (Running on oeis4.)