1,2

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

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

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

a(n)=sum(k*A110220(n, k), k=0..floor(n/2)). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 16 2005

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

a:=n->(1/n)*sum(k*binomial(n, k)*binomial(n+k, k+1), k=0..n): seq(a(n), n=1..22); (Deutsch)

Adjacent sequences: A025999 A026000 A026001 this_sequence A026003 A026004 A026005

Sequence in context: A144635 A097165 A152268 this_sequence A057009 A140480 A002315

nonn

Clark Kimberling (ck6(AT)evansville.edu)