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A108666 Number of (1,1)-steps in all Delannoy paths of length n (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)). 5
0, 1, 8, 57, 384, 2505, 16008, 100849, 628736, 3888657, 23900040, 146146473, 889928064, 5399971161, 32668236552, 197123362785, 1186790473728, 7131032334369, 42773183020296, 256161548120857, 1531966218561920 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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).

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

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->sum(k*binomial(n, k)*binomial(2*n-k, n), k=1..n): seq(a(n), n=0..24);

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

CROSSREFS

Cf. A001850, A104684.

Sequence in context: A244201 A079926 A283125 * A295711 A164031 A297369

Adjacent sequences:  A108663 A108664 A108665 * A108667 A108668 A108669

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Jul 07 2005

STATUS

approved

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Last modified February 16 08:26 EST 2019. Contains 320159 sequences. (Running on oeis4.)