1,5

T(n,k)=0 unless 1 <= k <= (n+1)/2.

The even-indexed rows have g.f. A(x, y):=Sum_{1<=k<=n}a(n, k)x^(2n)*y^k satisfying the functional equation A(x, y)(1+x*y^2) = x*y(1+(y+1)A(x, y+2)). The odd-indexed rows have g.f. B(x, y):=Sum_{1<=k<=n}b(n, k)x^(2n-1)*y^k satisfying the slightly different equation B(x, y)(1+x*(y+1)^2) = x*y(1+(y+1)B(x, y+2)). The recurrence relations underlying these functional equations are given in the Mathematica code below.

Table begins

\ k..1....2....3......

n

1 |..1

2 |..1

3 |..1 1

4 |..2....3

5 |..5....8....3

6 |.16...30...15

7 |.61..121...75...15

8 |272..588..420..105

For example, w = 21534 has 2 left-to-right maxima: w_1 = 2 and w_3 = 5.

T(4,2) = 3 because 2143, 3142, 3241 each have 2 left-to-right maxima.

Clear[a, b] EvenMultiplier[k_, j_]/; j<=k-2 := 0; EvenMultiplier[k_, j_]/; j>=k-1 := (2^(j+1-k) (Binomial[j, k-2]+Binomial[j+1, k-1])); a[1, 1]=1; a[n_, 0]:=0; a[n_, k_]/; 1<=k<=n && n>1 := a[n, k] = Sum[EvenMultiplier[k, j]a[n-1, j], {j, k-1, n-1}]; OddMultiplier[k_, j_]:=EvenMultiplier[k, j]-If[j==k-1, 2, 0]-If[j==k, 1, 0]; b[1, 1]=1; b[n_, 0]:=0; b[n_, k_]/; 1<=k<=n && n>1 := b[n, k] = Sum[OddMultiplier[k, j]b[n-1, j], {j, k-1, n-1}] Flatten[Table[{ Table[b[n, k], {k, n}], Table[a[n, k], {k, n}] }, {n, 7} ], 1]

Row sums are the up-down numbers (A000111), as is column k=1. Topmost entries in each column form the double factorials (A001147). The even indexed rows form A085734.

Adjacent sequences: A098903 A098904 A098905 this_sequence A098907 A098908 A098909

Sequence in context: A105955 A003893 A064737 this_sequence A007887 A105472 A030132

nonn,tabl

David Callan (callan(AT)stat.wisc.edu), Nov 04 2004