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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. 13
 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, 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, 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,10 LINKS Alois P. Heinz, Antidiagonals n = 0..140, flattened EXAMPLE A(6,0) = 3: [1,1,1,1,1,1], [1,2,3], [1,5]. A(5,1) = 4: [1,1,1,1,1], [1,1,1,2], [1,2,2], [1,4]. A(4,2) = 4: [1,1,1,1], [1,1,2], [1,2,1], [1,3]. Square array A(n,k) begins:   1,  1,  1,  1,  1,  1,  1,  1,  1, ...   1,  1,  1,  1,  1,  1,  1,  1,  1, ...   1,  1,  1,  1,  1,  1,  1,  1,  1, ...   2,  2,  2,  2,  2,  2,  2,  2,  2, ...   2,  3,  4,  4,  4,  4,  4,  4,  4, ...   2,  4,  7,  7,  8,  8,  8,  8,  8, ...   3,  6, 11, 13, 15, 15, 16, 16, 16, ...   2,  9, 18, 23, 28, 29, 31, 31, 32, ...   2, 13, 32, 40, 52, 56, 60, 61, 63, ... MAPLE b:= proc(n, i, j, k) option remember; `if`(n=0, 1,       `if`(i=0, add(b(n-h, j, h, k), h=1..n), add(        b(n-h, j, h, k), h=max(1, 2*j-i-k)..min(n, 2*j-i+k))))     end: A:= (n, k)-> `if`(n=0, 1, b(n-1, 0, 1, k)): seq(seq(A(n, d-n), n=0..d), d=0..12); MATHEMATICA 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 *) CROSSREFS Columns k=0-10 gives: A129654, A239551, A239552, A239553, A239554, A239555, A239556, A239557, A239558, A239559, A239560. Main diagonal gives A239561. Sequence in context: A276317 A289944 A055215 * A058398 A091499 A284249 Adjacent sequences:  A239547 A239548 A239549 * A239551 A239552 A239553 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Mar 21 2014 STATUS approved

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Last modified January 19 20:41 EST 2020. Contains 331066 sequences. (Running on oeis4.)