OFFSET
0,3
COMMENTS
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).
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..200 from Vincenzo Librandi)
Luca Ferrari and Emanuele Munarini, Enumeration of edges in some lattices of paths, arXiv:1203.6792 [math.CO], 2012 and J. Int. Seq. 17 (2014) #14.1.5
Robert A. Sulanke, Objects Counted by the Central Delannoy Numbers, Journal of Integer Sequences, Volume 6, 2003, Article 03.1.5.
FORMULA
a(n) = Sum_{k=0..n} k*A104684(k).
a(n) = Sum_{k=1..n} k*binomial(n, k)*binomial(2*n-k, n).
G.f.: x*(1-x)/(1-6*x+x^2)^(3/2).
D-finite with recurrence (n-1)*(2*n-3)*a(n) = 4*(3*n^2-6*n+2)*a(n-1) - (n-1)*(2*n-1)*a(n-2). - Vaclav Kotesovec, Oct 18 2012
a(n) ~ (3+2*sqrt(2))^n*sqrt(n)/(2^(7/4)*sqrt(Pi)). - Vaclav Kotesovec, Oct 18 2012
a(n) = n^2*hypergeom([-n+1, -n+1], [2], 2). - Peter Luschny, Jan 20 2020
EXAMPLE
a(2)=8 because in the 13 (=A001850(2)) Delannoy paths of length 2, namely, DD, DNE,DEN,NED,END,NDE,EDN,NENE,NEEN,ENNE,ENEN,NNEE and EENN, we have a total of eight D steps.
MAPLE
a := n -> add(k*binomial(n, k)*binomial(2*n-k, n), k=1..n): seq(a(n), n=0..24);
# Alternative:
a := n -> n^2*hypergeom([-n+1, -n+1], [2], 2):
seq(simplify(a(n)), n=0..24); # Peter Luschny, Jan 20 2020
MATHEMATICA
CoefficientList[Series[x*(1-x)/(1-6*x+x^2)^(3/2), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 18 2012 *)
PROG
(PARI) for(n=0, 25, print1(sum(k=0, n, k*binomial(n, k)*binomial(2*n-k, n)), ", ")) \\ G. C. Greubel, Jan 31 2017
(Python)
from math import comb
def A108666(n): return sum(comb(n, k)**2*k<<k-1 for k in range(1, n+1)) if n else 0 # Chai Wah Wu, Mar 22 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Jul 07 2005
STATUS
approved