cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A229892 Number T(n,k) of k up, k down permutations of [n]; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 0, 2, 1, 1, 0, 5, 3, 1, 1, 0, 16, 6, 4, 1, 1, 0, 61, 26, 10, 5, 1, 1, 0, 272, 71, 20, 15, 6, 1, 1, 0, 1385, 413, 125, 35, 21, 7, 1, 1, 0, 7936, 1456, 461, 70, 56, 28, 8, 1, 1, 0, 50521, 10576, 1301, 574, 126, 84, 36, 9, 1, 1
Offset: 0

Views

Author

Alois P. Heinz, Oct 02 2013

Keywords

Comments

T(n,k) is defined for n,k >= 0. The triangle contains only the terms with k<=n. T(n,k) = T(n,n) = A000012(n) = 1 for k>n.
T(2*n,n) = C(2*n-1,n) = A088218(n) = A001700(n-1) for n>0.
T(2*n+1,n) = C(2*n,n) = A000984(n).
T(2*n+1,n+1) = C(2n,n-1) = A001791(n) for n>0.

Examples

			Triangle T(n,k) begins:
  1;
  1,    1;
  0,    1,   1;
  0,    2,   1,   1;
  0,    5,   3,   1,  1;
  0,   16,   6,   4,  1,  1;
  0,   61,  26,  10,  5,  1, 1;
  0,  272,  71,  20, 15,  6, 1, 1;
  0, 1385, 413, 125, 35, 21, 7, 1, 1;

Links

Crossrefs

Columns k=1-10 give: A000111, A058258, A229884, A229885, A229886, A229887, A229888, A229889, A229890, A229891.
Cf. A227941, A229066, A229551.

Programs

  • Maple
    b:= proc(u, o, t, k) option remember; `if`(u+o=0, 1, add(`if`(t=k,
       b(o-j, u+j-1, 1, k), b(u+j-1, o-j, t+1, k)), j=1..o))
    end:
T:= (n, k)-> `if`(k+1>=n, 1, `if`(k=0, 0, b(0, n, 0, k))):
seq(seq(T(n, k), k=0..n), n=0..10);
  • Mathematica
    b[u_, o_, t_, k_] := b[u, o, t, k] = If[u+o == 0, 1, Sum[If[t == k, b[o-j, u+j-1, 1, k], b[u+j-1, o-j, t+1, k]], {j, 1, o}]]; t[n_, k_] := If[k+1 >= n, 1, If[k == 0, 0, b[0, n, 0, k]]]; Table[Table[t[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Dec 17 2013, translated from Maple *)

Formula

T(7,3) = 20: 1237654, 1247653, 1257643, 1267543, 1347652, 1357642, 1367542, 1457632, 1467532, 1567432, 2347651, 2357641, 2367541, 2457631, 2467531, 2567431, 3457621, 3467521, 3567421, 4567321.

A227941 Number of 1 up, 3 down, 5 up, 7 down, ... permutations of [n].

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 19, 55, 125, 245, 434, 4060, 21186, 81212, 254813, 692678, 1688555, 3776432, 60101767, 511650887, 3089821383, 14824989723, 60057570858, 213302293918, 681247718668, 1992449334436, 5409214694961, 132273848506202, 1692162553490943
Offset: 0

Views

Author

Alois P. Heinz, Oct 03 2013

Keywords

Examples

			a(2) = 1: 12.
a(3) = 2: 132, 231.
a(4) = 3: 1432, 2431, 3421.
a(5) = 4: 15432, 25431, 35421, 45321.
a(6) = 19: 154326, 164325, 165324, 165423, 254316, 264315, 265314, 265413, 354216, 364215, 365214, 365412, 453216, 463215, 465213, 465312, 563214, 564213, 564312.
a(7) = 55: 1543267, 1643257, ..., 6753124, 6754123.
a(8) = 125: 15432678, 16432578, ..., 78641235, 78651234.

Links

Crossrefs

Cf. A005408, A229066, A229551, A229892.

Programs

  • Maple
    b:= proc(u, o, t, k) option remember; `if`(u+o=0, 1, add(`if`(t=2*k-1,
       b(o-j, u+j-1, 1, k+1), b(u+j-1, o-j, t+1, k)), j=1..o))
    end:
a:= n-> b(0, n, 0, 1):
seq(a(n), n=0..35);
  • Mathematica
    b[u_, o_, t_, k_] := b[u, o, t, k] = If[u+o == 0, 1, Sum[If[t == 2*k-1, b[o-j, u+j-1, 1, k+1], b[u+j-1, o-j, t+1, k]], {j, 1, o}]];
a[n_] := b[0, n, 0, 1];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Sep 01 2022, after Alois P. Heinz *)

A229066 Number of 1 up, 2 down, 3 up, 4 down, ... permutations of [n].

Original entry on oeis.org

1, 1, 1, 2, 3, 11, 26, 50, 315, 1168, 3309, 7910, 78134, 431354, 1748956, 5797168, 16619603, 239887424, 1875375485, 10496708022, 47013492080, 178807998112, 599025922320, 11965846097382, 126883998286089, 947079890934441, 5574231845278396, 27500583638094490
Offset: 0

Views

Author

Alois P. Heinz, Oct 02 2013

Keywords

Examples

			a(2) = 1: 12.
a(3) = 2: 132, 231.
a(4) = 3: 1432, 2431, 3421.
a(5) = 11: 14325, 15324, 15423, 24315, 25314, 25413, 34215, 35214, 35412, 45213, 45312.
a(6) = 26: 143256, 153246, ..., 563124, 564123.
a(7) = 50: 1432567, 1532467, ..., 6741235, 6751234.
a(8) = 315: 14325687, 14325786, ..., 78613452, 78623451.

Links

Crossrefs

Cf. A227941, A229551, A229892.

Programs

  • Maple
    b:= proc(u, o, t, k) option remember; `if`(u+o=0, 1, add(`if`(t=k,
       b(o-j, u+j-1, 1, k+1), b(u+j-1, o-j, t+1, k)), j=1..o))
    end:
a:= n-> b(0, n, 0, 1):
seq(a(n), n=0..35);
  • Mathematica
    b[u_, o_, t_, k_] := b[u, o, t, k] = If[u + o == 0, 1, Sum[If[t == k, b[o - j, u + j - 1, 1, k + 1], b[u + j - 1, o - j, t + 1, k]], {j, 1, o}]];
a[n_] := b[0, n, 0, 1];
a /@ Range[0, 35] (* Jean-François Alcover, Mar 22 2021, after Alois P. Heinz *)

A247550 Number of 1 up, 1 down, 2 up, 1 down, 3 up, 1 down, ... permutations of [n].

Original entry on oeis.org

1, 1, 1, 2, 5, 9, 40, 169, 477, 1099, 8766, 56341, 234717, 774279, 2182270, 27260478, 249339033, 1457282467, 6624389780, 25274016620, 84400336507, 1518975557185, 18799199683021, 147690564521818, 892559422156897, 4474464873070564, 19410417198316364
Offset: 0

Views

Author

Alois P. Heinz, Sep 19 2014

Keywords

Examples

			a(5) = 9: 13245, 14235, 15234, 23145, 24135, 25134, 34125, 35124, 45123.
a(6) = 40: 132465, 132564, 142365, 142563, 143562, 152364, 152463, 153462, 162354, 162453, 163452, 231465, 231564, 241365, 241563, 243561, 251364, 251463, 253461, 261354, 261453, 263451, 341265, 341562, 342561, 351264, 351462, 352461, 361254, 361452, 362451, 451263, 451362, 452361, 461253, 461352, 462351, 561243, 561342, 562341.

Links

Crossrefs

Cf. A229551.

Programs

  • Maple
    b:= proc(u, o, t, k) option remember; `if`(u+o=0, 1, `if`(t>0,
       add(b(u+j-1, o-j, `if`(t=k, 0, t+1), k), j=1..o),
       add(b(u-j, o+j-1, 1, k+1), j=1..u)))
    end:
a:= n-> b(n, 0$3):
seq(a(n), n=0..35);
  • Mathematica
    b[u_, o_, t_, k_] := b[u, o, t, k] = If[u + o == 0, 1, If[t > 0,
     Sum[b[u + j - 1, o - j, If[t == k, 0, t + 1], k], {j, 1, o}],
     Sum[b[u - j, o + j - 1, 1, k + 1], {j, 1, u}]]];
a[n_] := b[n, 0, 0, 0];
a /@ Range[0, 35] (* Jean-François Alcover, Mar 26 2021, after Alois P. Heinz *)
Showing 1-4 of 4 results.

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