A362582 Triangular array read by rows. T(n,k) is the number of alternating permutations of [2n+1] having exactly 2k elements to the left of 1, n >= 0, 0 <= k <= n.

1, 1, 1, 5, 6, 5, 61, 75, 75, 61, 1385, 1708, 1750, 1708, 1385, 50521, 62325, 64050, 64050, 62325, 50521, 2702765, 3334386, 3427875, 3438204, 3427875, 3334386, 2702765, 199360981, 245951615, 252857605, 253708455, 253708455, 252857605, 245951615, 199360981

Offset: 0

Author

Geoffrey Critzer, Apr 25 2023

Keywords

• nonn
• tabl

Comments

Here, w = w_1,w_2,...,w_(2n+1) is an alternating permutation if w_1 < w_2 > w_3 < ... < w_(2n) > w_(2n+1).

Examples

			T(2,1) = 6 because we have: {2, 3, 1, 5, 4}, {2, 4, 1, 5, 3}, {2, 5, 1, 4, 3}, {3, 4, 1, 5, 2}, {3, 5, 1, 4, 2}, {4, 5, 1, 3, 2}.
Triangle begins
     1;
     1,     1;
     5,     6,     5;
    61,    75,    75,    61;
  1385,  1708,  1750,  1708,  1385;
 50521, 62325, 64050, 64050, 62325, 50521;
 ...
		

Links

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

Crossrefs

Cf. A000182 (row sums), A000364 (column k=0), A000680.

Cf. A000111, A104345.

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(2*n, 2*k)*b(2*k, 0)*b(2*(n-k), 0):
seq(seq(T(n, k), k=0..n), n=0..8);  # Alois P. Heinz, Apr 25 2023

Mathematica

nn = 6; B[n_] := (2 n)!/2^n; e[z_] := Sum[z^n/B[n], {n, 0, nn}]; Map[Select[#, # > 0 &] &,Table[B[n], {n, 0, nn}] CoefficientList[Series[1/e[-u z]*1/e[-z], {z, 0, nn}], {z, u}]] // Grid

Formula

Sum_{n>=0} Sum_{k=0..n} T(n,k)*u^k*z^n/A000680(n) = 1/(E(-u*z)*E(-z)) where E(z) = Sum_{n>=0} z^n/A000680(n).

T(n,k) = binomial(2*n,2*k)*A000111(2*k)*A000111(2*(n-k)). - Alois P. Heinz, Apr 25 2023