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Number of (1,1)-steps on the lines y=x, y=x+1 and y=x-1 in all Delannoy paths of length n.
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%I #14 Jul 24 2022 11:10:46

%S 0,1,8,55,354,2205,13484,81523,489158,2919481,17356752,102884271,

%T 608460330,3591886293,21172419444,124649246955,733107494286,

%U 4307974826097,25296523200920,148448166035239,870665283937010

%N Number of (1,1)-steps on the lines y=x, y=x+1 and y=x-1 in all Delannoy paths of length n.

%C A Delannoy path of length n is a path from (0,0) to (n,n), consisting of steps E=(1,0), N=(0,1) and D=(1,1).

%H Robert A. Sulanke, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL6/Sulanke/delannoy.html">Objects Counted by the Central Delannoy Numbers</a>, Journal of Integer Sequences, Volume 6, 2003, Article 03.1.5.

%F a(n) = sum(k*A110183(n,k),k=0..n).

%F G.f.: z(1-z-zr+z^2+z^2*r)/[(1-6z+z^2)(1-3z+z^2-zr+z^2*r)], where r=1+zr+zr^2=[1-z-sqrt(1-6z+z^2)]/(2z) is the g.f. of the large Schroeder numbers (A006318).

%F D-finite with recurrence n*a(n) +(-13*n+11)*a(n-1) +10*(5*n-9)*a(n-2) +10*(-5*n+16)*a(n-3) +(13*n-54)*a(n-4) +(-n+5)*a(n-5)=0. - _R. J. Mathar_, Jul 24 2022

%e a(3)=55 because on the 63 (=A001850(3)) Delannoy paths of length 3 we have altogether A108666(3)=57 D-steps; however 2 of these, namely the D's in NNDEE and EEDNN, are not on the lines y=x, y=x+1, y=x-1.

%p r:=(1-z-sqrt(1-6*z+z^2))/2/z: G:=z*(1-z-z*r+z^2+z^2*r)/(1-6*z+z^2)/(1-3*z+z^2-z*r+z^2*r): Gser:=series(G,z=0,27): 0,seq(coeff(Gser,z^n),n=1..24);

%Y Cf. A001850, A108666, A110183.

%K nonn

%O 0,3

%A _Emeric Deutsch_, Jul 14 2005