

A128740


Number of DD's in all skew Dyck paths of semilength n.


2



0, 0, 1, 6, 31, 154, 754, 3670, 17824, 86524, 420169, 2041946, 9932959, 48368000, 235769011, 1150413818, 5618786629, 27468246832, 134399280931, 658139933938, 3225323325109, 15817633139722, 77625378841756, 381190465089138
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OFFSET

0,4


COMMENTS

A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the xaxis, 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.


LINKS

Robert Israel, Table of n, a(n) for n = 0..1432
E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 21912203


FORMULA

a(n) = Sum_{k>=0} k*A128738(n,k).
G.f.: (2zg  g  z + 1)/(3zg  z  1), where g = 1 + zg^2 + z(g1) = (1  z  sqrt(1  6z + 5z^2))/(2z).
Conjecture: +2*(n+1)*(n^25*n+5)*a(n) +(13*n^3+60*n^252*n+15)*a(n1) +2*(8*n^344*n^2+63*n20)*a(n2) 5*(n3)*(n^23*n+1)*a(n3)=0.  R. J. Mathar, Jun 17 2016
Conjecture verified using the differential equation (12*z^2+10*z+10)*y(z)+(25*z^410*z^3+37*z^26*z)*y'(z)+(30*z^5+56*z^418*z^32*z^2)*y''(z)+(5*z^6+16*z^513*z^4+2*z^3)*y'''(z)+6*z^2=0 satisfied by the G.f..  Robert Israel, Dec 25 2017


EXAMPLE

a(3)=6 because each of the paths UDUUDD, UUDDUD, UUDUDD, UUUDDL contains one DD, the path UUUDDD contains 2 DD's and the paths UDUDUD, UDUUDL, UUUDLD, UUDUDL and UUUDLL contain no DD's.


MAPLE

g:=(1zsqrt(16*z+5*z^2))/2/z: ser:=series((2*z*ggz+1)/(3*z*gz1), z=0, 30): seq(coeff(ser, z, n), n=0..27);


CROSSREFS

Cf. A128738.
Sequence in context: A268401 A240879 A056015 * A227505 A026705 A243874
Adjacent sequences: A128737 A128738 A128739 * A128741 A128742 A128743


KEYWORD

nonn


AUTHOR

Emeric Deutsch, Mar 31 2007


STATUS

approved



