A287213 -id:A287213 - cp's OEIS frontend

A287215 Number T(n,k) of set partitions of [n] such that the maximal absolute difference between the least elements of consecutive blocks equals k; triangle T(n,k), n>=0, 0<=k<=max(n-1,0), read by rows.

1, 1, 1, 1, 1, 3, 1, 1, 8, 5, 1, 1, 22, 21, 7, 1, 1, 65, 86, 39, 11, 1, 1, 209, 361, 209, 77, 19, 1, 1, 732, 1584, 1123, 493, 171, 35, 1, 1, 2780, 7315, 6153, 3124, 1293, 413, 67, 1, 1, 11377, 35635, 34723, 20019, 9320, 3709, 1059, 131, 1, 1, 49863, 183080, 202852, 130916, 66992, 30396, 11373, 2837, 259, 1

Offset: 0

Alois P. Heinz, May 21 2017

The maximal absolute difference is assumed to be zero if there are fewer than two blocks.

T(n,k) is defined for all n,k >= 0. The triangle contains only the positive terms. T(n,k) = 0 if k>=n and k>0.

			T(4,0) = 1: 1234.
T(4,1) = 8: 134|2, 13|24, 14|23, 1|234, 14|2|3, 1|24|3, 1|2|34, 1|2|3|4.
T(4,2) = 5: 124|3, 12|34, 12|3|4, 13|2|4, 1|23|4.
T(4,3) = 1: 123|4.
Triangle T(n,k) begins:
  1;
  1;
  1,   1;
  1,   3,    1;
  1,   8,    5,    1;
  1,  22,   21,    7,   1;
  1,  65,   86,   39,  11,   1;
  1, 209,  361,  209,  77,  19,  1;
  1, 732, 1584, 1123, 493, 171, 35, 1;

Alois P. Heinz, Rows n = 0..141, flattened
Wikipedia, Partition of a set

Columns k=0-10 give: A000012, A003101(n-1), A322875, A322876, A322877, A322878, A322879, A322880, A322881, A322882, A322883.
Row sums give A000110.
T(2n,n) gives A322884.
Cf. A287213, A287216, A287416, A287640.

b:= proc(n, k, m, l) option remember; `if`(n<1, 1,
     `if`(l-n>k, 0, b(n-1, k, m+1, n))+m*b(n-1, k, m, l))
    end:
A:= (n, k)-> b(n-1, min(k, n-1), 1, n):
T:= (n, k)-> A(n, k)-`if`(k=0, 0, A(n, k-1)):
seq(seq(T(n, k), k=0..max(n-1, 0)), n=0..12);

b[n_, k_, m_, l_] := b[n, k, m, l] = If[n < 1, 1, If[l - n > k, 0, b[n - 1, k, m + 1, n]] + m*b[n - 1, k, m, l]];
A[n_, k_] := b[n - 1, Min[k, n - 1], 1, n];
T[n_, k_] := A[n, k] - If[k == 0, 0, A[n, k - 1]];
Table[T[n, k], {n, 0, 12}, {k, 0, Max[n - 1, 0]}] // Flatten (* Jean-François Alcover, May 19 2018, after Alois P. Heinz *)

T(n,k) = A287216(n,k) - A287216(n,k-1) for k>0, T(n,0) = 1.

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.

Original entry on oeis.org

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

Offset: 0

Alois P. Heinz, May 21 2017

The sequence of column k satisfies a linear recurrence with constant coefficients of order 2^(k-1) for k>0.

			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, ...

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

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.

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);

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 *)

A(n,k) = Sum_{j=0..k} A287213(n,j).

A258109 Number of balanced parenthesis expressions of length 2n and depth 3.

Original entry on oeis.org

1, 5, 18, 57, 169, 482, 1341, 3669, 9922, 26609, 70929, 188226, 497845, 1313501, 3459042, 9096393, 23895673, 62721698, 164531565, 431397285, 1130708866, 2962826465, 7761964833, 20331456642, 53249182309, 139449644717, 365166860706, 956185155129, 2503657040137

Offset: 3

Gheorghe Coserea, May 20 2015

a(n) is the number of Dyck paths of length 2n and height 3. For example, a(3) = 1 because there is only one such Dyck path which is UUUDDD. - Ran Pan, Sep 26 2015

a(n) is the number of rooted plane trees with n+1 nodes and height 3 (see example for correspondence). - Gheorghe Coserea, Feb 04 2016

			For n=4, the a(4) = 5 solutions are
                /\       /\
               /  \        \
LRLLLRRR    /\/    \        \
................................
                /\        |
             /\/  \      / \
LLRLLRRR    /      \        \
................................
              /\/\        |
             /    \       |
LLLRLRRR    /      \     / \
................................
              /\          |
             /  \/\      / \
LLLRRLRR    /      \    /
................................
              /\          /\
             /  \        /
LLLRRRLR    /    \/\    /

S. S. Skiena and M. A. Revilla, Programming Challenges: The Programming Contest Training Manual, Springer, 2006, page 140.

Alois P. Heinz, Table of n, a(n) for n = 3..1000 (first 40 terms from Gheorghe Coserea)
Kyu-Hwan Lee, Se-jin Oh, Catalan triangle numbers and binomial coefficients, arXiv:1601.06685 [math.CO], 2016.
Index entries for linear recurrences with constant coefficients, signature (5,-7,2).

Column k=3 of A080936.
Column k=2 of A287213.
Cf. A000045, A000079, A262600.

typedef long long unsigned Integer;
Integer a(int n)
{
    int i;
    Integer pow2 = 1, a[3] = {0};
    for (i = 3; i <= n; ++i) {
        a[ i%3 ] = pow2 + 3 * a[ (i-1)%3 ] - a[ (i-2)%3 ];
        pow2 = pow2 * 2;
    }
    return a[ (i-1)%3 ];
}

I:=[1,5,18,57,169]; [n le 5 select I[n] else 5*Self(n-1) - 7*Self(n-2) + 2*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Sep 26 2015

a:= proc(n) option remember; `if`(n<3, 0,
      `if`(n=3, 1, 2^(n-3) +3*a(n-1) -a(n-2)))
    end:
seq(a(n), n=3..30);  # Alois P. Heinz, May 20 2015

Join[{1, 5}, LinearRecurrence[{5, -7, 2}, {18, 57, 169}, 30]] (* Vincenzo Librandi, Sep 26 2015 *)

Vec(-x^3/((2*x-1)*(x^2-3*x+1)) + O(x^100)) \\ Colin Barker, May 24 2015

a(n) = fibonacci(2*n-1) - 2^(n-1)  \\ Gheorghe Coserea, Feb 04 2016

a(n) = 2^(n-3) + 3 * a(n-1) - a(n-2).
a(n) = 5*a(n-1) - 7*a(n-2) + 2*a(n-3) for n>5. - Colin Barker, May 24 2015
G.f.: -x^3 / ((2*x-1)*(x^2-3*x+1)). - Colin Barker, May 24 2015
a(n) = A000045(2n-1) - A000079(n-1). - Gheorghe Coserea, Feb 04 2016
a(n) = 2^(-1-n)*(-5*4^n - (-5+sqrt(5))*(3+sqrt(5))^n + (3-sqrt(5))^n*(5+sqrt(5))) / 5. - Colin Barker, Jun 05 2017
a(n) = Sum_{i=1..n-1} A061667(i)*(n-1-i) - Tim C. Flowers, May 16 2018

A287416 Number T(n,k) of set partitions of [n] such that the maximal value of all absolute differences between least elements of consecutive blocks and between consecutive elements within the blocks equals k; triangle T(n,k), n>=0, 0<=k<=max(n-1,0), read by rows.

Original entry on oeis.org

1, 1, 0, 2, 0, 3, 2, 0, 4, 8, 3, 0, 5, 22, 19, 6, 0, 6, 52, 81, 48, 16, 0, 7, 114, 289, 267, 147, 53, 0, 8, 240, 941, 1250, 968, 529, 204, 0, 9, 494, 2894, 5310, 5469, 3919, 2174, 878, 0, 10, 1004, 8601, 21256, 28083, 25326, 17593, 9961, 4141

Offset: 0

Alois P. Heinz, May 24 2017

The maximal value is assumed to be zero if there are no consecutive blocks and no consecutive elements.

T(n,k) is defined for all n,k >= 0. The triangle contains only the terms for k <= max(n-1,0). T(n,k) = 0 if k>=n and k>0.

			T(4,1) = 4: 1234, 1|234, 1|2|34, 1|2|3|4.
T(4,2) = 8: 124|3, 12|34, 12|3|4, 134|2, 13|24, 13|2|4, 1|23|4, 1|24|3.
T(4,3) = 3: 123|4, 14|23, 14|2|3.
T(5,3) = 19: 1235|4, 123|45, 123|4|5, 125|34, 125|3|4, 134|25, 134|2|5, 13|24|5, 13|25|4, 145|23, 14|235, 14|23|5, 1|234|5, 145|2|3, 14|25|3, 14|2|35, 14|2|3|5, 1|25|34, 1|25|3|4.
Triangle T(n,k) begins:
  1;
  1;
  0, 2;
  0, 3,   2;
  0, 4,   8,   3;
  0, 5,  22,  19,    6;
  0, 6,  52,  81,   48,  16;
  0, 7, 114, 289,  267, 147,  53;
  0, 8, 240, 941, 1250, 968, 529, 204;
  ...

Alois P. Heinz, Rows n = 0..23, flattened
Wikipedia, Partition of a set

Columns k=1-2 give: A001477 (for n>1), A005803 (for n>0).
Row sums give A000110.
Cf. A287213, A287215, A287417, A287640.

b:= proc(n, k, l, t) option remember; `if`(n<1, 1, `if`(t-n>k, 0,
       b(n-1, k, map(x-> `if`(x-n>=k, [][], x), [l[], n]), n)) +add(
       b(n-1, k, sort(map(x-> `if`(x-n>=k, [][], x), subsop(j=n, l))),
       `if`(t-n>k, infinity, t)), j=1..nops(l)))
    end:
A:= (n, k)-> b(n, min(k, n-1), [], n):
T:= (n, k)-> A(n, k)-`if`(k=0, 0, A(n, k-1)):
seq(seq(T(n, k), k=0..max(n-1, 0)), n=0..12);

b[n_, k_, l_, t_] := b[n, k, l, t] = If[n < 1, 1, If[t - n > k, 0, b[n - 1, k, If[# - n >= k, Nothing, #]& /@ Append[l, n], n]] + Sum[b[n - 1, k, Sort[If[# - n >= k, Nothing, #]& /@ ReplacePart[l, j -> n]], If[t - n > k, Infinity, t]], {j, 1, Length[l]}]];
A[n_, k_] := b[n, Min[k, n - 1], {}, n];
T[n_, k_] := A[n, k] - If[k == 0, 0, A[n, k - 1]];
Table[T[n, k], {n, 0, 12}, { k, 0, Max[n - 1, 0]}] // Flatten (* Jean-François Alcover, May 24 2018, translated from Maple *)

T(n,k) = A287417(n,k) - A287417(n,k-1) for k>0, T(n,0) = 1.

T(n+2,n+1) = 1 + A000110(n).

A287640 Number T(n,k) of set partitions of [n], where k is minimal such that for all j in [n]: j is member of block b implies b = 1 or at least one of j-1, ..., j-k is member of a block >= b-1; triangle T(n,k), n >= 0, 0 <= k <= max(floor(n/2), n-2), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 4, 1, 13, 1, 1, 41, 9, 1, 1, 131, 59, 11, 1, 1, 428, 344, 88, 15, 1, 1, 1429, 1906, 634, 146, 23, 1, 1, 4861, 10345, 4389, 1231, 280, 39, 1, 1, 16795, 55901, 30006, 9835, 2763, 602, 71, 1, 1, 58785, 303661, 205420, 77178, 25014, 6967, 1408, 135, 1

Offset: 0

Alois P. Heinz, May 28 2017

			T(4,0) = 1: 1234.
T(4,1) = 13: 123|4, 124|3, 12|34, 12|3|4, 134|2, 13|24, 14|23, 1|234, 1|23|4, 14|2|3, 1|24|3, 1|2|34, 1|2|3|4.
T(4,2) = 1: 13|2|4.
T(5,2) = 9: 124|3|5, 135|2|4, 13|25|4, 13|2|45, 13|2|4|5, 14|23|5, 14|2|35, 14|2|3|5, 1|24|3|5.
T(6,3) = 11: 1245|3|6, 1346|2|5, 134|26|5, 134|2|56, 134|2|5|6, 145|23|6, 145|2|36, 145|2|3|6, 14|25|3|6, 15|24|3|6, 1|245|3|6.
T(6,4) = 1: 1345|2|6.
T(7,4) = 15: 12456|3|7, 13457|2|6, 1345|27|6, 1345|2|67, 1345|2|6|7, 1456|23|7, 1456|2|37, 1456|2|3|7, 145|26|3|7, 146|25|3|7, 14|256|3|7, 156|24|3|7, 15|246|3|7, 16|245|3|7, 1|2456|3|7.
Triangle T(n,k) begins:
  1;
  1;
  1,    1;
  1,    4;
  1,   13,     1;
  1,   41,     9,    1;
  1,  131,    59,   11,    1;
  1,  428,   344,   88,   15,   1;
  1, 1429,  1906,  634,  146,  23,  1;
  1, 4861, 10345, 4389, 1231, 280, 39, 1;
  ...

Alois P. Heinz, Rows n = 0..20, flattened
Wikipedia, Partition of a set

Columns k=0-1 give: A000012, A001453.
Row sums give A000110.
Cf. A168415, A287213, A287215, A287416, A287641.

b:= proc(n, l) option remember; `if`(n=0 or l=[], 1, add(b(n-1,
      [seq(max(l[i], j), i=2..nops(l)), j]), j=1..l[1]+1))
    end:
T:= (n, k)-> `if`(k=0, 1, b(n, [0$k])-b(n, [0$k-1])):
seq(seq(T(n, k), k=0..max(n/2, n-2)), n=0..12);

b[n_, l_] := b[n, l] = If[n == 0 || l == {}, 1, Sum[b[n-1, Append[Table[ Max[l[[i]], j], {i, 2, Length[l]}], j]], {j, 1, l[[1]] + 1}]];
T[n_, k_] := If[k == 0, 1, b[n, Table[0, k]] - b[n, Table[0, k - 1]]];
Table[T[n, k], {n, 0, 12}, { k, 0, Max[n/2, n - 2]}] // Flatten (* Jean-François Alcover, May 22 2018, translated from Maple *)

T(n,k) = A287641(n,k) - A287641(n,k-1) for k>0, T(n,0) = 1.

T(n+4,n+1) = A168415(n) for n>0.

A294052 Number of set partitions of [n] such that the maximal absolute difference between consecutive elements within a block equals three.

Original entry on oeis.org

2, 13, 61, 248, 935, 3368, 11777, 40347, 136214, 454922, 1507000, 4961100, 16253188, 53045703, 172607505, 560317916, 1815445901, 5873136282, 18976870985, 61256217631, 197573796328, 636837047532, 2051636248268, 6606758265032, 21268025275930, 68445465415825

Offset: 4

Alois P. Heinz, Oct 22 2017

Alois P. Heinz, Table of n, a(n) for n = 4..1000
Index entries for linear recurrences with constant coefficients, signature (7,-15,8,5,-5,1)

Column k=3 of A287213.

Cf. A001519, A287275.

LinearRecurrence[{7,-15,8,5,-5,1},{2,13,61,248,935,3368},30] (* Harvey P. Dale, Nov 22 2023 *)

G.f.: (x-2)*x^4/((x-1)*(x^2-3*x+1)*(x^3-x^2-3*x+1)).

a(n) = A287275(n) - A001519(n).

A294053 Number of set partitions of [n] such that the maximal absolute difference between consecutive elements within a block equals four.

Original entry on oeis.org

5, 38, 215, 1061, 4835, 20973, 88010, 360787, 1453978, 5784863, 22790024, 89092968, 346161413, 1338360327, 5153828402, 19781784669, 75723483993, 289218958150, 1102597884045, 4196961350447, 15954736073286, 60585891849501, 229855881578197, 871373727460242

Offset: 5

Alois P. Heinz, Oct 22 2017

Alois P. Heinz, Table of n, a(n) for n = 5..1000
Index entries for linear recurrences with constant coefficients, signature (9,-26,23,0,20,-40,2,16,-1,0,-3,1)

Column k=4 of A287213.

Cf. A287275, A287276.

Drop[CoefficientList[Series[(x^5-2x^4+x^3-3x^2+7x-5)x^5/((x-1)(x^3-x^2- 3x+1)(x^8-x^7-7x^5+7x^4+x^3+4x^2-5x+1)),{x,0,30}],x],5] (* or *) LinearRecurrence[{9,-26,23,0,20,-40,2,16,-1,0,-3,1},{5,38,215,1061,4835,20973,88010,360787,1453978,5784863,22790024,89092968},30] (* Harvey P. Dale, May 25 2022 *)

G.f.: (x^5-2*x^4+x^3-3*x^2+7*x-5)*x^5 / ((x-1) *(x^3-x^2-3*x+1) *(x^8 -x^7 -7*x^5 +7*x^4 +x^3 +4*x^2 -5*x+1)).

a(n) = A287276(n) - A287275(n).

A294054 Number of set partitions of [n] such that the maximal absolute difference between consecutive elements within a block equals five.

Original entry on oeis.org

15, 129, 836, 4789, 25430, 128360, 625905, 2976800, 13896153, 63950894, 291050996, 1312981604, 5881158250, 26191105884, 116085151862, 512487018089, 2255036961813, 9895020092839, 43316960894877, 189247864529166, 825397574526671, 3594688070523059, 15635607417594050

Offset: 6

Alois P. Heinz, Oct 22 2017

Alois P. Heinz, Table of n, a(n) for n = 6..1000
Index entries for linear recurrences with constant coefficients, signature (11, -40, 49, -8, 14, 94, -354, 92, 41, 464, -113, -280, -39, -164, 217, 104, -48, -16, -32, 6, 5, 2, 1, -1)

Column k=5 of A287213.

Cf. A287276, A287277.

G.f.: (x^15-3*x^13-2*x^12-4*x^11+7*x^10+21*x^9+9*x^8-33*x^7-44*x^6+48*x^5-10*x^4 +18*x^3+17*x^2-36*x+15)*x^6 / ((x^8-x^7-7*x^5+7*x^4+x^3+4*x^2-5*x+1) *(x^10-x^9 -x^7 -9*x^6 +10*x^5+9*x^4-7*x^3+4*x^2-5*x+1)*(x^6+x^5-x^4-3*x^2-x+1)).

a(n) = A287277(n) - A287276(n).

A294055 Number of set partitions of [n] such that the maximal absolute difference between consecutive elements within a block equals six.

Original entry on oeis.org

52, 495, 3573, 22986, 138140, 791355, 4378359, 23619010, 124985133, 651528571, 3356054226, 17122003113, 86672668175, 435922323548, 2180749707230, 10860449611842, 53881317215173, 266455227999826, 1314042578677412, 6464913802646613, 31741357154688581, 155566738708385012

Offset: 7

Alois P. Heinz, Oct 22 2017

Alois P. Heinz, Table of n, a(n) for n = 7..1000

Column k=6 of A287213.

Cf. A287277, A287278.

a(n) = A287278(n) - A287277(n).

A294056 Number of set partitions of [n] such that the maximal absolute difference between consecutive elements within a block equals seven.

Original entry on oeis.org

203, 2108, 16657, 117897, 783922, 4993579, 30785899, 185088952, 1091673665, 6342173645, 36398304558, 206815079839, 1165446105293, 6522388312404, 36291251200978, 200938029161451, 1107900649341124, 6086624218751214, 33335270870796943, 182080367445331011, 992214076623396046

Offset: 8

Alois P. Heinz, Oct 22 2017

Alois P. Heinz, Table of n, a(n) for n = 8..1000

Column k=7 of A287213.

Cf. A287278, A287279.

a(n) = A287279(n) - A287278(n).

cp's OEIS Frontend

A287215 Number T(n,k) of set partitions of [n] such that the maximal absolute difference between the least elements of consecutive blocks equals k; triangle T(n,k), n>=0, 0<=k<=max(n-1,0), read by rows.

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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.

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Maple

Mathematica

Formula

A258109 Number of balanced parenthesis expressions of length 2n and depth 3.

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C

Magma

Maple

Mathematica

PARI

PARI

Formula

A287416 Number T(n,k) of set partitions of [n] such that the maximal value of all absolute differences between least elements of consecutive blocks and between consecutive elements within the blocks equals k; triangle T(n,k), n>=0, 0<=k<=max(n-1,0), read by rows.

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Maple

Mathematica

Formula

A287640 Number T(n,k) of set partitions of [n], where k is minimal such that for all j in [n]: j is member of block b implies b = 1 or at least one of j-1, ..., j-k is member of a block >= b-1; triangle T(n,k), n >= 0, 0 <= k <= max(floor(n/2), n-2), read by rows.

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Maple

Mathematica

Formula

A294052 Number of set partitions of [n] such that the maximal absolute difference between consecutive elements within a block equals three.

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Mathematica

Formula

A294053 Number of set partitions of [n] such that the maximal absolute difference between consecutive elements within a block equals four.

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