A108666 Number of (1,1)-steps in all Delannoy paths of length n.

0, 1, 8, 57, 384, 2505, 16008, 100849, 628736, 3888657, 23900040, 146146473, 889928064, 5399971161, 32668236552, 197123362785, 1186790473728, 7131032334369, 42773183020296, 256161548120857, 1531966218561920, 9150330147133161, 54591847064667528, 325361790187810257

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

a(n)/n = A047781(n) (for n >= 1).

Cf. A001850, A104684.

Sequence in context: A079926 A346228 A283125 * A295711 A164031 A297369

Adjacent sequences: A108663 A108664 A108665 * A108667 A108668 A108669

Keyword

nonn,changed

Author

Emeric Deutsch, Jul 07 2005

Status

approved