The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A108666 Number of (1, 1)-steps in all Delannoy paths of length n. 9
 0, 1, 8, 57, 384, 2505, 16008, 100849, 628736, 3888657, 23900040, 146146473, 889928064, 5399971161, 32668236552, 197123362785, 1186790473728, 7131032334369, 42773183020296, 256161548120857, 1531966218561920, 9150330147133161, 54591847064667528, 325361790187810257 (list; graph; refs; listen; history; text; internal format)
 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). 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). - 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): 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 CROSSREFS a(n)/n = A047781(n) (for n >= 1). 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified May 24 18:00 EDT 2020. Contains 334574 sequences. (Running on oeis4.)