%I A026002
%S A026002 1,7,41,231,1289,7183,40081,224143,1256465,7059735,39753273,224298231,
%T A026002 1267854873,7178461215,40704778785,231128079903,1314016698401,7478998203943,
%U A026002 42612705597769,243025194476551,1387226559025961,7924982285747247
%N A026002 a(n) = T(n,n+2), where T = Delannoy triangle (A008288).
%C A026002 Number of U steps in all lattice paths from (0,0) to (2n,0) consisting 
               of steps U=(1,1), D=(1,-1), H=(2,0) and never going below the x-axis 
               (i.e. Schroeder paths). For example, a(2)=7, counting the U's in 
               HH, UDUD, UUDD, UHD, HUD and UDH. - Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               Dec 06 2003
%C A026002 Number of UH's in all lattice paths from (0,0) to (2n+2,0) consisting 
               of steps U=(1,1), D=(1,-1), H=(2,0) and never going below the x-axis 
               (i.e. Schroeder paths). For example, a(2)=7, counting the UH's, shown 
               between parentheses, in the 22 (=A006318(3)) Schroeder paths of length 
               6: HHH, HHUD, HUDH, HUDUD, H(UH)D, HUUDD, (UH)DH, (UH)DUD, UUDDH, 
               UUDDUD, (UH)HD, (UH)UDD, UUDHD, UUDUDD, U(UH)DD, UUUDDD, UDHH, UDHUD, 
               UDUDH, UDUDUD, UD(UH)D and UDUUDD. - Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               Jul 16 2005
%F A026002 a(n)=(1/n)sum(k binomial(n, k)binomial(n+k, k+1), k=0..n). G.f.=1/2-1/
               (2z)+(1-4*z+z^2)/[2z sqrt(1-6z+z^2)]. - Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               Dec 06 2003
%F A026002 a(n)=sum(k*A110220(n, k), k=0..floor(n/2)). - Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               Jul 16 2005
%F A026002 a(n)=sum{k=0..n, C(n, k)C(n+2, k)2^k}; a(n)=Jacobi_P(n, 2, 0, 3); - Paul 
               Barry (pbarry(AT)wit.ie), Jan 23 2006
%p A026002 a:=n->(1/n)*sum(k*binomial(n,k)*binomial(n+k,k+1),k=0..n): seq(a(n),n=1..22); 
               (Deutsch)
%Y A026002 Sequence in context: A144635 A097165 A152268 this_sequence A057009 A140480 
               A002315
%Y A026002 Adjacent sequences: A025999 A026000 A026001 this_sequence A026003 A026004 
               A026005
%K A026002 nonn
%O A026002 1,2
%A A026002 Clark Kimberling (ck6(AT)evansville.edu)