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 A287214 Number A(n,k) of set partitions of [n] such that for each block all absolute differences between consecutive elements are <= k; square array A(n,k), n>=0, k>=0, read by antidiagonals. 13
 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 4, 1, 1, 1, 2, 5, 8, 1, 1, 1, 2, 5, 13, 16, 1, 1, 1, 2, 5, 15, 34, 32, 1, 1, 1, 2, 5, 15, 47, 89, 64, 1, 1, 1, 2, 5, 15, 52, 150, 233, 128, 1, 1, 1, 2, 5, 15, 52, 188, 481, 610, 256, 1, 1, 1, 2, 5, 15, 52, 203, 696, 1545, 1597, 512, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,9 COMMENTS The sequence of column k satisfies a linear recurrence with constant coefficients of order 2^(k-1) for k>0. LINKS Alois P. Heinz, Antidiagonals n = 0..45, flattened Pierpaolo Natalini, Paolo Emilio Ricci, New Bell-Sheffer Polynomial Sets, Axioms 2018, 7(4), 71. Wikipedia, Partition of a set FORMULA A(n,k) = Sum_{j=0..k} A287213(n,j). EXAMPLE A(4,0) = 1: 1|2|3|4. A(4,1) = 8: 1234, 123|4, 12|34, 12|3|4, 1|234, 1|23|4, 1|2|34, 1|2|3|4. A(4,2) = 13: 1234, 123|4, 124|3, 12|34, 12|3|4, 134|2, 13|24, 13|2|4, 1|234, 1|23|4, 1|24|3, 1|2|34, 1|2|3|4. Square array A(n,k) begins:   1,  1,   1,   1,   1,   1,   1,   1, ...   1,  1,   1,   1,   1,   1,   1,   1, ...   1,  2,   2,   2,   2,   2,   2,   2, ...   1,  4,   5,   5,   5,   5,   5,   5, ...   1,  8,  13,  15,  15,  15,  15,  15, ...   1, 16,  34,  47,  52,  52,  52,  52, ...   1, 32,  89, 150, 188, 203, 203, 203, ...   1, 64, 233, 481, 696, 825, 877, 877, ... MAPLE b:= proc(n, k, l) option remember; `if`(n=0, 1, b(n-1, k, map(x->       `if`(x-n>=k, [][], x), [l[], n]))+add(b(n-1, k, sort(map(x->       `if`(x-n>=k, [][], x), subsop(j=n, l)))), j=1..nops(l)))     end: A:= (n, k)-> b(n, min(k, n-1), []): seq(seq(A(n, d-n), n=0..d), d=0..12); MATHEMATICA b[0, _, _] = 1; b[n_, k_, l_List] := b[n, k, l] = b[n - 1, k, If[# - n >= k, Nothing, #]& /@ Append[l, n]] + Sum[b[n - 1, k, Sort[If[# - n >= k, Nothing, #]& /@ ReplacePart[l, j -> n]]], {j, 1, Length[l]}]; A[n_, k_] := b[n, Min[k, n - 1], {}]; Table[A[n, d - n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Apr 30 2018, after Alois P. Heinz *) CROSSREFS Columns k=0-10 give: A000012, A011782, A001519, A287275, A287276, A287277, A287278, A287279, A287280, A287281, A287282. Main diagonal gives A000110. Cf. A287213, A287216, A287417, A287641. Sequence in context: A030424 A216656 A295679 * A287216 A145515 A267383 Adjacent sequences:  A287211 A287212 A287213 * A287215 A287216 A287217 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, May 21 2017 STATUS approved

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Last modified August 9 20:24 EDT 2020. Contains 336326 sequences. (Running on oeis4.)