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%I #16 Dec 26 2017 03:22:44
%S 0,0,1,6,31,154,754,3670,17824,86524,420169,2041946,9932959,48368000,
%T 235769011,1150413818,5618786629,27468246832,134399280931,
%U 658139933938,3225323325109,15817633139722,77625378841756,381190465089138
%N Number of DD's in all skew Dyck paths of semilength n.
%C 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.
%H Robert Israel, <a href="/A128740/b128740.txt">Table of n, a(n) for n = 0..1432</a>
%H E. Deutsch, E. Munarini, S. Rinaldi, <a href="http://dx.doi.org/10.1016/j.jspi.2010.01.015">Skew Dyck paths</a>, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203
%F a(n) = Sum_{k>=0} k*A128738(n,k).
%F 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).
%F 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
%F 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
%e 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.
%p 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);
%Y Cf. A128738.
%K nonn
%O 0,4
%A _Emeric Deutsch_, Mar 31 2007
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