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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. 14
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

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.

LINKS

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

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.

EXAMPLE

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;

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-Fran├žois Alcover, Dec 17 2013, translated from Maple *)

CROSSREFS

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

Cf. A227941, A229066, A229551.

Sequence in context: A198237 A122049 A238802 * A064879 A173591 A156603

Adjacent sequences:  A229889 A229890 A229891 * A229893 A229894 A229895

KEYWORD

nonn,tabl,eigen

AUTHOR

Alois P. Heinz, Oct 02 2013

STATUS

approved

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Last modified December 9 13:50 EST 2019. Contains 329877 sequences. (Running on oeis4.)