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A239550 Number A(n,k) of compositions of n such that the first part is 1 and the second differences of the parts are in {-k,...,k}; square array A(n,k), n>=0, k>=0, read by antidiagonals.

%I A239550 #21 Oct 26 2018 22:32:34
%S A239550 1,1,1,1,1,1,1,1,1,2,1,1,1,2,2,1,1,1,2,3,2,1,1,1,2,4,4,3,1,1,1,2,4,7,
%T A239550 6,2,1,1,1,2,4,7,11,9,2,1,1,1,2,4,8,13,18,13,3,1,1,1,2,4,8,15,23,32,
%U A239550 18,3,1,1,1,2,4,8,15,28,40,53,24,2,1,1,1,2,4,8,16,29,52,73,89,34,3
%N A239550 Number A(n,k) of compositions of n such that the first part is 1 and the second differences of the parts are in {-k,...,k}; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%H A239550 Alois P. Heinz, <a href="/A239550/b239550.txt">Antidiagonals n = 0..140, flattened</a>
%e A239550 A(6,0) = 3: [1,1,1,1,1,1], [1,2,3], [1,5].
%e A239550 A(5,1) = 4: [1,1,1,1,1], [1,1,1,2], [1,2,2], [1,4].
%e A239550 A(4,2) = 4: [1,1,1,1], [1,1,2], [1,2,1], [1,3].
%e A239550 Square array A(n,k) begins:
%e A239550   1,  1,  1,  1,  1,  1,  1,  1,  1, ...
%e A239550   1,  1,  1,  1,  1,  1,  1,  1,  1, ...
%e A239550   1,  1,  1,  1,  1,  1,  1,  1,  1, ...
%e A239550   2,  2,  2,  2,  2,  2,  2,  2,  2, ...
%e A239550   2,  3,  4,  4,  4,  4,  4,  4,  4, ...
%e A239550   2,  4,  7,  7,  8,  8,  8,  8,  8, ...
%e A239550   3,  6, 11, 13, 15, 15, 16, 16, 16, ...
%e A239550   2,  9, 18, 23, 28, 29, 31, 31, 32, ...
%e A239550   2, 13, 32, 40, 52, 56, 60, 61, 63, ...
%p A239550 b:= proc(n, i, j, k) option remember; `if`(n=0, 1,
%p A239550       `if`(i=0, add(b(n-h, j, h, k), h=1..n), add(
%p A239550        b(n-h, j, h, k), h=max(1, 2*j-i-k)..min(n, 2*j-i+k))))
%p A239550     end:
%p A239550 A:= (n, k)-> `if`(n=0, 1, b(n-1, 0, 1, k)):
%p A239550 seq(seq(A(n, d-n), n=0..d), d=0..12);
%t A239550 b[n_, i_, j_, k_] := b[n, i, j, k] = If[n == 0, 1, If[i == 0, Sum[b[n-h, j, h, k], {h, 1, n}], Sum[b[n-h, j, h, k], {h, Max[1, 2*j - i - k], Min[n, 2*j - i + k]}]]] ; A[n_, k_] := If[n == 0, 1, b[n-1, 0, 1, k]]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* _Jean-François Alcover_, Jan 22 2015, after _Alois P. Heinz_ *)
%Y A239550 Columns k=0-10 gives: A129654, A239551, A239552, A239553, A239554, A239555, A239556, A239557, A239558, A239559, A239560.
%Y A239550 Main diagonal gives A239561.
%K A239550 nonn,tabl
%O A239550 0,10
%A A239550 _Alois P. Heinz_, Mar 21 2014