A033820 Triangle read by rows: T(k,j) = ((2*j+1)/(k+1))*binomial(2*j,j)*binomial(2*k-2*j,k-j).

1, 1, 3, 2, 4, 10, 5, 9, 15, 35, 14, 24, 36, 56, 126, 42, 70, 100, 140, 210, 462, 132, 216, 300, 400, 540, 792, 1716, 429, 693, 945, 1225, 1575, 2079, 3003, 6435, 1430, 2288, 3080, 3920, 4900, 6160, 8008, 11440, 24310, 4862, 7722, 10296, 12936, 15876, 19404

Offset: 0

Author

Alexander Burstein

Keywords

• nonn
• tabl

Comments

f(n,k) = 2^{n-2(k-2)}sum(T(k-2,j)*binomial(n+2*(k-2-j),2*(k-2-j)),j=0..k-2) is the number of length n k-ary strings (k >= 2) which avoid a rising triple (pattern 123) or any other given 3-letter permutation pattern.

Row sums are the powers of 4. This is explained by a simple statistic on the 4^n lattice paths of length 2n formed from upsteps U=(1,1) and downsteps D=(1,-1). For such a path, define X = number of upsteps that lie above ground level (GL), the horizontal line through the initial vertex, and before the last vertex at GL. For UDDUUUUDDU for instance, the last vertex at GL follows the fourth step, and so X = 1. T(n,k) is the number of these paths with X=n-k. For example, T(2,1)=4 counts UDUU, UDDU, UDDD, DUUD because each has n-k=1 upsteps above GL and before the last vertex at GL. - David Callan, Nov 21 2011

Examples

			{1},
{1, 3},
{2, 4, 10},
{5, 9, 15, 35},
{14, 24, 36, 56, 126},
{42, 70, 100, 140, 210, 462},
{132, 216, 300, 400, 540, 792, 1716},
...
		

Links

• Alexander Burstein, Enumeration of words with forbidden patterns, Ph.D. thesis, University of Pennsylvania, 1998.
• Ira Gessel, Super ballot numbers, J. Symbolic Computation 14 (1992), 179-194.
• Walter Shur, Two Game-Set Inequalities, J. Integer Seqs., Vol. 6, 2003.

Crossrefs

Cf. A000108, A000984, A000302.

Essentially a reflected version of A078817.

Formula

T(k,0) = binomial(2*k, k)/(k+1), the k-th Catalan number; T(k,k) = binomial(2*(k+1),k+1)/2, half the (k+1)-st central binomial coefficient sum of entries in row k (over j) = 2^{2*(k-1)}

T(k,j) = sum(C(k-i)D(i), i=0..j), C(i) = binomial(2*i,i)/(i+1), D(i) = binomial(2*i,i).

G.f.: 2/(1-4*x*y+sqrt((1-4*x)*(1-4*x*y))). - Vladeta Jovovic, Dec 14 2003

Extensions

More terms from Vladeta Jovovic, Dec 10 2003