A128781 Triangle of numbers a(n,k), n>=3, ceiling((n-3)/2)<=k<=n-3: a(n,k)=Sum[ Binomial[x + y + z, x]*Binomial[y + z, y]*Binomial[n - 2 - x - 2*y - 2*z, 2*n - 2*y - 5 - 2*k]*(2^x)*((-1)^z), {z, 0, (2*k - n + 3)/2}, {y, 0, n - 3 - k}, {x, 0, 2*k - n + 3 - 2*z}].

1, 4, 2, 10, 12, 20, 3, 42, 35, 24, 112, 56, 4, 108, 252, 84, 40, 360, 504, 120, 5, 220, 990, 924, 165, 60, 880, 2376, 1584, 220, 6, 390, 2860, 5148, 2574, 286, 84, 1820, 8008, 10296, 4004, 364, 7, 630, 6825, 20020, 19305, 6006, 455, 112, 3360, 21840, 45760

Offset

3,2

Comments

a(n,k) counts the permutations in S_n which have zero occurrences of the pattern 213 and one occurrence of the pattern 123 and k descents.

References

D. Hök, Parvisa mönster i permutationer [Swedish], (2007).

Links

Alois P. Heinz, Rows n = 3..201, flattened

Formula

a(n,k) = s(n,k)+t(n,k), s(n,k) = a(n-1,k-1), t(n,k) = C(n-2,2*n-5-2*k) + t(n-1,k-1) + s(n-1,k), a(3,0)=t(3,0)=1.

Example

Triangle begins:
n\k  0   1   2    3    4    5   6
----------------------------------
3    1;
4    .   4;
5    .   .  10;
6    .   .  12,  20;
7    .   .   3,  42,  35;
8    .   .   .   24, 112,  56;
9    .   .   .    4, 108, 252, 84;

Crossrefs

Diagonal gives A000292.

Sequence in context: A210735 A075086 A284782 * A135440 A215500 A188128

Adjacent sequences: A128778 A128779 A128780 * A128782 A128783 A128784

Keyword

nonn,tabf

Author

David Hoek (david.hok(AT)telia.com), Mar 28 2007

Extensions

Edited by Peter Bala, Dec 05 2013

Status

approved