%I A114121
%S A114121 1,2,7,26,99,382,1486,5812,22819,89846,354522,1401292,5546382,21977516,
%T A114121 87167164,345994216,1374282019,5461770406,21717436834,86392108636,
%U A114121 343801171354,1368640564996,5450095992964,21708901408216,86492546019214
%N A114121 Expansion of (sqrt(1-4x)+(1-2x)/(2(1-4x)).
%C A114121 Second binomial transform of A032443 with interpolated zeros.
%C A114121 a(n) = total number of lattice points, taken over all Dyck n-paths (A000108), 
               that (i) lie on or above ground level and (ii) lie on or directly 
               below a peak. For example with n=2, UUDD has 1 peak contributing 
               3 lattice points--(2,0), (2,1) and (2,2) when the path starts at 
               the origin--and UDUD has 2 peaks, each contributing 2 lattice points 
               and so a(2)=3+4=7. - David Callan (callan(AT)stat.wisc.edu), Jul 
               14 2006
%C A114121 Hankel transform is binomial(n+2,2). - Paul Barry (pbarry(AT)wit.ie), 
               Dec 04 2007
%C A114121 Image of (-1)^n under the Riordan array ((1/2)(1/(1-4x)+1/sqrt(1-4x)),
               c(x)-1), c(x) the g.f. of A000108. - Paul Barry (pbarry(AT)wit.ie), 
               Jun 15 2008
%F A114121 a(n)=sum{k=0..n, C(n,k)*2^(n-k-2)*(2^k+C(k,k/2))(1+(-1)^k)}; a(n)=(A000984(n)+A081294(n))/
               2.
%F A114121 G.f.: (1-4x+(1-2x)sqrt(1-4x))/(2(1-4x)^(3/2)); a(n)=sum{k=0..n, sum{j=0..n, 
               C(2n,n-k-j)(-1)^j}}; - Paul Barry (pbarry(AT)wit.ie), Jun 15 2008
%F A114121 a(n)=sum{k=0..n, C(2n,n-k)*(1+(-1)^k)/2}. [From Paul Barry (pbarry(AT)wit.ie), 
               Aug 06 2009]
%F A114121 Contribution from Paul Barry (pbarry(AT)wit.ie), Sep 07 2009: (Start)
%F A114121 a(n)=C(2n-1,n-1)+(4^n+3*0^n)/4.
%F A114121 Integral representation a(n)=(1/(2*pi))*Int(x^n/sqrt(x(4-x)),x,0,4)+(4^n+0^n)/
               4. (End)
%Y A114121 Sequence in context: A001075 A113436 A126223 this_sequence A049775 A101850 
               A045868
%Y A114121 Adjacent sequences: A114118 A114119 A114120 this_sequence A114122 A114123 
               A114124
%K A114121 easy,nonn
%O A114121 0,2
%A A114121 Paul Barry (pbarry(AT)wit.ie), Feb 13 2006