0,2

Convolution of A002212 with itself.

Number of skew Dyck paths of semilength n+1 starting with UU. A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1)(up), D=(1,-1)(down) and L=(-1,-1)(left) so that up and left steps do not overlap. The length of the path is defined to be the number of its steps. Example: a(2)=7 because we have UUDDUD, UUDUDD, UUDUDL, UUUDDD, UUUDDL, UUUDLD and UUUDLL. - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 11 2007

S. J. Cyvin et al., Enumeration and classification of certain polygonal systems...: annelated catafusenes, J. Chem. Inform. Comput. Sci., 34 (1994), 1174-1180.

E. Deutsch, E. Munarini and S. Rinaldi, Skew Dyck paths (in preparation).

a(n)=(2/n)*sum(binomial(n, j)*binomial(2j+1, j-1), j=1..n) for n>=1.

(n+2)*a(n) = (6*n+2)*a(n-1)-(5*n-10)*a(n-2). - Vladeta Jovovic (vladeta(AT)eunet.rs), Jul 16 2004

a := n->(2/n)*sum(binomial(n, j)*binomial(2*j+1, j-1), j=1..n): 1, seq(a(n), n=1..22);

(PARI) a(n)=polcoeff((1-x-sqrt(1-6*x+5*x^2+x^2*O(x^n)))^2/4, n+2)

T(n, n-1) where T is A055450.

Essentially the first differences of A002212 and A025238.

Sequence in context: A114121 A049775 A101850 this_sequence A129482 A150536 A150537

Adjacent sequences: A045865 A045866 A045867 this_sequence A045869 A045870 A045871

nonn

N. J. A. Sloane (njas(AT)research.att.com).

More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), May 11 2007