%I A033820
%S A033820 1,1,3,2,4,10,5,9,15,35,14,24,36,56,126,42,70,100,140,210,462,132,216,
%T A033820 300,400,540,792,1716,429,693,945,1225,1575,2079,3003,6435,1430,2288,
%U A033820 3080,3920,4900,6160,8008,11440,24310,4862,7722,10296,12936,15876,19404
%N A033820 Triangle read by rows: T(k,j) = ((2*j+1)/(k+1))*binomial(2*j,j)*binomial(2*k-2*j,
               k-j).
%C A033820 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.
%H A033820 Alexander Burstein, <a href="http://www.math.upenn.edu/~alexb/mythesis.dvi">
               Enumeration of words with forbidden patterns</a>, Ph.D. thesis, University 
               of Pennsylvania, 1998.
%H A033820 Walter Shur, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               Two Game-Set Inequalities</a>, J. Integer Seqs., Vol. 6, 2003.
%H A033820 Ira Gessel, <a href="http://www.cs.brandeis.edu/~ira/">Super ballot numbers</
               a>.
%F A033820 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)}
%F A033820 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).
%F A033820 G.f.: 2/(1-4*x*y+sqrt((1-4*x)*(1-4*x*y))). - Vladeta Jovovic (vladeta(AT)eunet.rs), 
               Dec 14 2003
%e A033820 {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}, ...
%Y A033820 Cf. A000108, A000984, A000302.
%Y A033820 Essentially a reflected version of A078817.
%Y A033820 Sequence in context: A083164 A094962 A084793 this_sequence A095259 A137824 
               A019321
%Y A033820 Adjacent sequences: A033817 A033818 A033819 this_sequence A033821 A033822 
               A033823
%K A033820 nonn,tabl
%O A033820 0,3
%A A033820 Alexander Burstein (alexb(AT)math.upenn.edu)
%E A033820 More terms from Vladeta Jovovic (vladeta(AT)eunet.rs), Dec 10 2003