1,2

Number of Schroeder paths (i.e. consisting of steps U=(1,1), D=(1,-1), and H=(2,0) and never going below the x-axis) from (0,0) to (2n+2,0), with exactly one peak at an even level. E.g. a(2)=6 because we have UUDDH, HUUDD, UDUUDD, UUDDUD, UUDHD, and UHUDD. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 28 2003

Number of left steps in all skew Dyck paths of semilength n+1. 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)=6 because in the 10 (=A002212(3)) skew Dyck paths of semilength 3 ( namely UDUUDL, UUUDLD, UUDUDL, UUUDDL, UUUDLL, and five Dyck paths that have no left steps) we have altogether 6 left steps. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 05 2007

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

E.g.f.: exp(3x) I_1(2x), where I_1 is Bessel function. - Michael Somos, Sep 09 2002.

G.f.: [1-3z-sqrt(1-6z+5z^2)]/[2zsqrt(1-6z+5z^2)] - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 25 2003

a(n)=[t^(n+1)](1+3t+t^2)^n. a := n->sum(3^(2j-n-1)*binomial(n, j)*binomial(j, n+1-j), j=ceil((n+1)/2)..n). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 30 2004

a(n)=sum{k=0..n, binomial(n, k)binomial(2k, k+1)} - Paul Barry (pbarry(AT)wit.ie), Sep 20 2004

a(n)=n*A002212(n). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 05 2007

(PARI) a(n)=if(n<0, 0, polcoeff((1+3*x+x^2)^n, n-1))

Cf. A006318.

Cf. A002212.

Adjacent sequences: A026373 A026374 A026375 this_sequence A026377 A026378 A026379

Sequence in context: A003279 A082134 A030192 this_sequence A026899 A135160 A046945

nonn

Clark Kimberling (ck6(AT)evansville.edu)