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

%I A108666 #42 Jun 15 2025 14:16:44
%S A108666 0,1,8,57,384,2505,16008,100849,628736,3888657,23900040,146146473,
%T A108666 889928064,5399971161,32668236552,197123362785,1186790473728,
%U A108666 7131032334369,42773183020296,256161548120857,1531966218561920,9150330147133161,54591847064667528,325361790187810257
%N A108666 Number of (1,1)-steps in all Delannoy paths of length n.
%C A108666 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).
%H A108666 G. C. Greubel, <a href="/A108666/b108666.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..200 from Vincenzo Librandi)
%H A108666 Luca Ferrari and Emanuele Munarini, <a href="http://arxiv.org/abs/1203.6792">Enumeration of edges in some lattices of paths</a>, arXiv:1203.6792 [math.CO], 2012 and <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Ferrari/ferrari.html">J. Int. Seq. 17 (2014) #14.1.5</a>
%H A108666 Robert A. Sulanke, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL6/Sulanke/delannoy.html">Objects Counted by the Central Delannoy Numbers</a>, Journal of Integer Sequences, Volume 6, 2003, Article 03.1.5.
%F A108666 a(n) = Sum_{k=0..n} k*A104684(k).
%F A108666 a(n) = Sum_{k=1..n} k*binomial(n, k)*binomial(2*n-k, n).
%F A108666 G.f.: x*(1-x)/(1-6*x+x^2)^(3/2).
%F A108666 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
%F A108666 a(n) ~ (3+2*sqrt(2))^n*sqrt(n)/(2^(7/4)*sqrt(Pi)). - _Vaclav Kotesovec_, Oct 18 2012
%F A108666 a(n) = n^2*hypergeom([-n+1, -n+1], [2], 2). - _Peter Luschny_, Jan 20 2020
%F A108666 a(n) = Sum_{k=1..n} 2^(k-1)*k*binomial(n,k)^2. - _Ridouane Oudra_, Jun 15 2025
%e A108666 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.
%p A108666 a := n -> add(k*binomial(n,k)*binomial(2*n-k,n),k=1..n): seq(a(n),n=0..24);
%p A108666 # Alternative:
%p A108666 a := n -> n^2*hypergeom([-n+1, -n+1], [2], 2):
%p A108666 seq(simplify(a(n)), n=0..24); # _Peter Luschny_, Jan 20 2020
%t A108666 CoefficientList[Series[x*(1-x)/(1-6*x+x^2)^(3/2), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Oct 18 2012 *)
%o A108666 (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
%o A108666 (Python)
%o A108666 from math import comb
%o A108666 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
%Y A108666 a(n)/n = A047781(n) (for n >= 1).
%Y A108666 Cf. A001850, A104684.
%K A108666 nonn
%O A108666 0,3
%A A108666 _Emeric Deutsch_, Jul 07 2005