A212206 Irregular triangle read by rows: T(n,k) = number of "pat" permutations of [1..n] with k descents.

1, 0, 1, 0, 2, 0, 1, 4, 0, 0, 12, 2, 0, 0, 12, 30, 0, 0, 4, 100, 28, 0, 0, 0, 140, 280, 9, 0, 0, 0, 90, 980, 360, 0, 0, 0, 22, 1680, 2940, 220, 0, 0, 0, 0, 1540, 10584, 4620, 52, 0, 0, 0, 0, 728, 20790, 33264, 4004, 0, 0, 0, 0, 140, 24024, 121968, 60060, 1820, 0, 0, 0, 0, 0, 16380, 264264, 396396, 65520, 340, 0, 0, 0, 0, 0, 6120

Offset

1,5

Comments

Row sums are Catalan numbers A000108.

Links

Table of n, a(n) for n=1..81.

D. Callan, Flexagons lead to a Catalan number identity, Amer. Math. Monthly, 119 (May 2012), 415-419.

Tad White, Quota Trees, arXiv:2401.01462 [math.CO], 2024. See p. 20.

Formula

T(n,k) = binomial(2n-2k-1,k)*binomial(2k,n-k-1)/(2n-2k-1).

Example

Triangle begins:
1
0 1
0 2
0 1 4
0 0 12 2
0 0 12 30
0 0 4 100 28
0 0 0 140 280 9
0 0 0 90 980 360
0 0 0 22 1680 2940 220
...

Maple

A212206 := proc(n, k)
        binomial(2*n-2*k-1, k)*binomial(2*k, n-k-1)/(2*n-2*k-1) ;
end proc:
for n from 0 to 15 do
        for k from 0 to floor((2*n-1)/3) do
                printf("%d, ", A212206(n, k)) ;
        end do:
end do; # R. J. Mathar, Jun 26 2012

Crossrefs

Cf. A000108.

Sequence in context: A099089 A121298 A372873 * A247489 A208756 A259873

Adjacent sequences: A212203 A212204 A212205 * A212207 A212208 A212209

Keyword

nonn,tabf

Author

N. J. A. Sloane, May 15 2012

Status

approved