A110184 Number of (1,1)-steps on the lines y=x, y=x+1 and y=x-1 in all Delannoy paths of length n.
0, 1, 8, 55, 354, 2205, 13484, 81523, 489158, 2919481, 17356752, 102884271, 608460330, 3591886293, 21172419444, 124649246955, 733107494286, 4307974826097, 25296523200920, 148448166035239, 870665283937010
Offset: 0
Keywords
Examples
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.
Links
- Robert A. Sulanke, Objects Counted by the Central Delannoy Numbers, Journal of Integer Sequences, Volume 6, 2003, Article 03.1.5.
Programs
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Maple
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);
Formula
a(n) = sum(k*A110183(n,k),k=0..n).
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).
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
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