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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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

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.

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) 2191-2203

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(g-1) = (1 - z - sqrt(1 - 6z + 5z^2))/(2z).

Conjecture: +2*(n+1)*(n^2-5*n+5)*a(n) +(-13*n^3+60*n^2-52*n+15)*a(n-1) +2*(8*n^3-44*n^2+63*n-20)*a(n-2) -5*(n-3)*(n^2-3*n+1)*a(n-3)=0. - R. J. Mathar, Jun 17 2016

Conjecture verified using the differential equation (-12*z^2+10*z+10)*y(z)+(-25*z^4-10*z^3+37*z^2-6*z)*y'(z)+(-30*z^5+56*z^4-18*z^3-2*z^2)*y''(z)+(-5*z^6+16*z^5-13*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:=(1-z-sqrt(1-6*z+5*z^2))/2/z: ser:=series((2*z*g-g-z+1)/(3*z*g-z-1), 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

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Last modified January 22 04:27 EST 2020. Contains 331133 sequences. (Running on oeis4.)