A104345 Triangle read by rows: T(n,k) is the number of alternating permutations on [n+1] with 1 in position k+1, 0<=k<=n.

1, 1, 1, 1, 2, 1, 2, 3, 3, 2, 5, 8, 6, 8, 5, 16, 25, 20, 20, 25, 16, 61, 96, 75, 80, 75, 96, 61, 272, 427, 336, 350, 350, 336, 427, 272, 1385, 2176, 1708, 1792, 1750, 1792, 1708, 2176, 1385, 7936, 12465, 9792, 10248, 10080, 10080, 10248, 9792, 12465, 7936

Offset: 0

Author

David Callan, Mar 02 2005

Keywords

• nonn
• tabl

Examples

			Table begins
\ k..0....1....2....3....4....
n
0 |..1
1 |..1....1
2 |..1....2....1
3 |..2....3....3....2
4 |..5....8....6....8....5
5 |.16...25...20...20...25...16
6 |.61...96...75...80...75...96...61
7 |272..427..336..350..350..336..427..272
For example, T(3,1) counts 2143, 3142, 4132 - the alternating permutations on [4] with 1 in position 2.
		

Links

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

Crossrefs

Cf. A104346. Row sums are A001250; column k=0 and main diagonal are the up-down numbers (A000111); column k=1 is A065619.

T(2n,n) gives A362581.

Programs

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, 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, 0]*b[n-k, 0];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Apr 01 2024, after Alois P. Heinz *)

Formula

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

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