A104346 Triangle read by rows: T(n,k) is the number of alternating max-precedes-min permutations on [n+2] with 1 in position k+2, 0<=k<=n.

1, 1, 1, 1, 2, 2, 2, 3, 6, 5, 5, 8, 12, 20, 16, 16, 25, 40, 50, 80, 61, 61, 96, 150, 200, 240, 366, 272, 272, 427, 672, 875, 1120, 1281, 1904, 1385, 1385, 2176, 3416, 4480, 5600, 6832, 7616, 11080, 7936, 7936, 12465, 19584, 25620, 32256, 38430, 45696, 49860, 71424, 50521

Offset

0,5

Links

Alois P. Heinz, Rows n = 0..150, flattened

Formula

The mixed e.g.f./o.g.f. is Sum_{k=0..n} T(n, k)*x^n/n!*y^k = (sec(x) + tan(x))*sec(xy)*(sec(xy) + tan(xy)).

T(n,k) = binomial(n,k)*A000111(k+1)*A000111(n-k). - Alois P. Heinz, Apr 25 2023

Example

Table begins
\ k..0....1....2....3....4....
n
0 |..1
1 |..1....1
2 |..1....2....2
3 |..2....3....6....5
4 |..5....8...12...20...16
5 |.16...25...40...50...80...61
6 |.61...96..150..200..240..366..272
For example, a(3,1) counts 45132, 35142, 25143---the alternating permutations on [5] with 5 preceding 1 and 1 in position 3.

Maple

b:= proc(u, o) option remember; `if`(u+o=0, 1,
      add(b(o-1+j, u-j), j=1..u))
    end:
T:= (n, k)-> binomial(n, k)*b(k+1, 0)*b(n-k, 0):
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Apr 25 2023

Mathematica

b[u_, o_] := b[u, o] = If[u+o == 0, 1, Sum[b[o-1+j, u-j], {j, 1, u}]];
T[n_, k_] := Binomial[n, k]*b[k+1, 0]*b[n-k, 0];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Apr 21 2025, after Alois P. Heinz *)

Crossrefs

Cf. A104345. Row sums, column k=0 and main diagonal are all A000111.

Sequence in context: A378277 A070610 A156820 * A318238 A341075 A193450

Adjacent sequences: A104343 A104344 A104345 * A104347 A104348 A104349

Keyword

nonn,tabl

Author

David Callan, Mar 02 2005

Status

approved