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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
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
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: A346226 A240879 A056015 * A227505 A026705 A243874
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Mar 31 2007
STATUS
approved