A229884 -id:A229884 - cp's OEIS frontend

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.

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

Alois P. Heinz, Oct 02 2013

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.

			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;

Alois P. Heinz, Rows n = 0..140, flattened

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

Cf. A227941, A229066, A229551.

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);

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 *)

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

A005982 3 up, 3 down, 3 up, ... permutations of length 3n+1.

Original entry on oeis.org

1, 20, 1301, 202840, 61889101, 32676403052, 27418828825961, 34361404413755056, 61335081309931829401, 150221740688275657957940, 489799709605132718770274141, 2073641570051429601078643837960, 11163099186064084100687107863253381

Offset: 1

N. J. A. Sloane

nonn

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
P. R. Stein, personal communication.

Alois P. Heinz, Table of n, a(n) for n = 1..100
P. R. Stein & N. J. A. Sloane, Correspondence, 1975

Cf. A229884.

b:= proc(u, o, t) option remember; `if`(u+o=0, 1, add(`if`(t=3,
       b(o-j, u+j-1, 1), b(u+j-1, o-j, t+1)), j=1..o))
    end:
a:= n-> b(0, 3*n+1, 0):
seq(a(n), n=1..20);  # Alois P. Heinz, Oct 06 2013

b[u_, o_, t_] := b[u, o, t] = If[u+o == 0, 1, Sum[If[t == 3, b[o-j, u+j-1, 1], b[u+j-1, o-j, t+1]], {j, 1, o}]] ; a[n_] := b[0, 3*n+1, 0]; Table[a[n], {n, 1, 20}] (* Jean-François Alcover, Nov 25 2014, after Alois P. Heinz *)

Typo in name fixed by Alois P. Heinz, Oct 06 2013

cp's OEIS Frontend

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.

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