%I A059435
%S A059435 1,2,12,88,720,6304,57792,547712,5323008,52761088,531311616,5420488704,
%T A059435 55905767424,581954543616,6106210615296,64513688174592,685741070942208,
%U A059435 7328106153115648,78684992821788672,848487859401261056
%N A059435 Number of lattice paths in plane starting at (0,0) and ending at (n,n) 
               with steps from {(i,j):i+j>0,i,j >= 0} that never never go below 
               the line y=x.
%C A059435 Series reversion of x(1-4x)/(1-2x). - Paul Barry (pbarry(AT)wit.ie), 
               May 19 2005
%C A059435 The Hankel transform of this sequence is 8^C(n+1,2)= [1,8,512,262144,
               ...] . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 08 2007
%D A059435 Ira M. Gessel, A factorization for formal Laurent series and lattice 
               path enumeration, J. Combin. Theory Ser. A 28 (1980), 321-337.
%D A059435 W.-J. Woan, A bijective proof by induction that the n-th term of this 
               sequence is 2^(n-1) times of the n-th term of the big Schroeder number, 
               Jan 28, 2001. (unpublished)
%H A059435 Robert A. Sulanke, Counting Lattice Paths by Narayana Polynomials, <a 
               href="http://www.combinatorics.org/">Electronic J. Combinatorics 
               </a>, Vol. 7, R40, 2000.
%H A059435 David Callan, A uniformly distributed statistic on a class of lattice 
               paths, <a href="http://www.combinatorics.org/">Electronic J. Combinatorics</
               a>, Vol. 11(1), R82, 2004.
%F A059435 [1+2x-sqrt(4x^2-12x+1)]/8x
%F A059435 a(n)=sum{k=0..n, C(n+1, k)C(2n-k, n)(-1)^k*4^(n-k)*2^k}/(n+1); a(n)=sum{k=0..n, 
               (1/n)*C(n, k)*C(n, k+1)*4^k*2^(n-k)}; a(n)=sum{k=0..n, N(n, k)*4^k*2^(n-k)}, 
               N(n, k) Narayana numbers (A001263). - Paul Barry (pbarry(AT)wit.ie), 
               May 19 2005
%p A059435 gf := (1+2*x-sqrt(4*x^2-12*x+1))/(8*x): s := series(gf, x, 100): for 
               i from 0 to 50 do printf(`%d,`,coeff(s,x,i)) od:
%Y A059435 A006318, A001003.
%Y A059435 a(n) = 2^(n-1)*A001003(n-1).
%Y A059435 Cf. A054726.
%Y A059435 a(n) = 2^n*A001003(n).
%Y A059435 Sequence in context: A052867 A097237 A055531 this_sequence A143923 A079858 
               A121357
%Y A059435 Adjacent sequences: A059432 A059433 A059434 this_sequence A059436 A059437 
               A059438
%K A059435 nonn,easy
%O A059435 0,2
%A A059435 Wen-jin Woan (wwoan(AT)fac.howard.edu), Feb 01 2001
%E A059435 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Feb 01 2001