%I A110440
%S A110440 1,3,1,11,6,1,45,31,9,1,197,156,60,12,1,903,785,360,98,15,1,4279,3978,
%T A110440 2061,684,145,18,1,20793,20335,11529,4403,1155,201,21,1,103049,104856,
%U A110440 63728,27048,8270,1800,266,24,1,518859,545073,350136,161412,55458,14202
%N A110440 Triangular array formed by the little Schroeder numbers. s(n,k)= the 
               number of unit step restricted paths (i.e. they never go below the 
               x-axis) from the origin (0,0) to (n-1,k-1) using up step U(1,1), 
               three types of level steps L(1,0),L'(1,0),L"(1,0) and two types of 
               down steps D(1,-1),D'(1,-1). s(0,0)=1 and the leftmost column s(n,
               0) is A001003.
%C A110440 This sequence factors A038255 into a product of Riordan arrays.
%D A110440 Naiomi T. Cameron and Asamoah Nkwanta, On Some (Pseudo) Involutions in 
               the Riordan Group, Journal of Integer Sequences, Vol. 8 (2005), Article 
               05.3.7.
%F A110440 Recurrence is s(n+1, 0)= 3s(n, k)+ 2s(n, k+1) for the leftmost column 
               entries and s(n+1, k)= s(n, k-1)+ 3s(n, k)+ 2s(n, k+1) for the other 
               column entries. Riordan array ((1-3z-sqrt(1-6z+z^2))/4z*z, (1-3z-sqrt(1-6z+z^2))/
               4z)
%F A110440 Sum_{k, k>=0} T(m, k)*T(n, k)*2^k = T(m+n, 0) = A001003(m+n+1) . - Philippe 
               DELEHAM (kolotoko(AT)wanadoo.fr), Sep 14 2005
%F A110440 G.f.: 2/( 1 -x*L -2*x*y*U + sqrt( (1 -x*L)^2 -4*x^2*D*U ) ) where L=3, 
               U=1, D=2. - Michael Somos Mar 31 2007
%e A110440 Triangle starts:
%e A110440 1;
%e A110440 3,1;
%e A110440 11,6,1;
%e A110440 45,31,9,1;
%e A110440 197,156,60,12,1; ...
%o A110440 (PARI) {T(n, k)= if(n<0| k>n, 0, polcoeff(polcoeff( 2/(1 -3*x -2*x*y 
               +sqrt( 1 -6*x +x^2 +x*O(x^n)) ), n), k))} /* Michael Somos Mar 31 
               2007 */
%Y A110440 Sequence in context: A113955 A110165 A111965 this_sequence A135574 A008969 
               A119908
%Y A110440 Adjacent sequences: A110437 A110438 A110439 this_sequence A110441 A110442 
               A110443
%K A110440 easy,nice,nonn,tabl
%O A110440 0,2
%A A110440 Asamoah Nkwanta (nkwanta(AT)jewel.morgan.edu), Aug 08 2005