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%I #32 Jun 05 2021 16:41:34
%S 1,2,8,36,172,852,4324,22332,116876,618084,3296308,17702412,95627580,
%T 519170004,2830862532,15494401116,85091200620,468692890308,
%U 2588521289812,14330490031020,79509491551772,442019710668852
%N Number of Delannoy paths of length n with no (1,1)-steps on the line y=x.
%C 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).
%C Equals left border of triangle A152250 and INVERTi transform of A001850, the Delannoy numbers: (1, 3, 13, 63, 321, ...). - _Gary W. Adamson_, Nov 30 2008
%C Hankel transform is A036442. First column of Riordan array ((1-x)/(1+x), x/(1+3x+2x^2))^{-1}. - _Paul Barry_, Apr 27 2009
%H Vincenzo Librandi, <a href="/A109980/b109980.txt">Table of n, a(n) for n = 0..200</a>
%H Robert A. Sulanke, <a href="https://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 G.f.: 1/(z + sqrt(1 - 6*z + z^2)).
%F Moment representation: a(n) = 0^n/3 + (1/Pi)*Integral_{x=3-2*sqrt(2)..3+2*sqrt(2)} x^n*sqrt(-x^2+6x-1)/(x*(6-x)) dx. - _Paul Barry_, Apr 27 2009
%F From _Gary W. Adamson_, Aug 23 2011: (Start)
%F a(n) is the upper left term in M^n, M = an infinite square production matrix as follows:
%F 2, 2, 0, 0, 0, 0, ...
%F 2, 1, 2, 0, 0, 0, ...
%F 2, 1, 1, 2, 0, 0, ...
%F 2, 1, 1, 1, 2, 0, ...
%F 2, 1, 1, 1, 1, 2, ...
%F ... (End)
%F D-finite with recurrence: n*a(n) = 3*(4*n-3)*a(n-1) - (37*n-57)*a(n-2) + 6*(n-3)*a(n-3). - _Vaclav Kotesovec_, Oct 18 2012
%F a(n) ~ 2^(1/4) * (1 + sqrt(2))^(2*n+3) / (sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Oct 18 2012, simplified Dec 24 2017
%e a(2)=8 because we have NDE, EDN, NENE, NEEN, ENNE, ENEN, NNEE and EENN.
%p g:=1/(z+sqrt(1-6*z+z^2)): gser:=series(g,z=0,28): 1,seq(coeff(gser,z^n),n=1..25);
%t CoefficientList[Series[1/(x+Sqrt[1-6*x+x^2]), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Oct 18 2012 *)
%Y First column of A109979.
%Y Cf. A152250.
%K nonn
%O 0,2
%A _Emeric Deutsch_, Jul 06 2005
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