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%I #54 Apr 19 2024 11:30:11
%S 1,2,10,50,258,1362,7306,39650,217090,1196834,6634890,36949266,
%T 206549250,1158337650,6513914634,36718533570,207412854786,
%U 1173779487810,6653482333450,37770112857074,214694383882498,1221832400430482,6961037946938250,39697830840765090,226596964146630658
%N First differences of the central Delannoy numbers (A001850).
%C Number of Delannoy paths of length n that do not start with a (1, 1) step (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)). Example: a(1) = 2 because we have NE and EN. Column 0 of A110169 (also nonzero entries in each column of A110169).
%C For n > 0: a(n) = A128966(2*n,n). - _Reinhard Zumkeller_, Jul 20 2013
%H Vincenzo Librandi, <a href="/A110170/b110170.txt">Table of n, a(n) for n = 0..200</a>
%H Thomas Selig, <a href="https://arxiv.org/abs/2202.06487">Combinatorial aspects of sandpile models on wheel and fan graphs</a>, arXiv:2202.06487 [math.CO], 2022.
%H 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 G.f.: (1-z)/sqrt(1-6*z+z^2).
%F a(n) = P_n(3) - P_{n-1}(3) (n >= 1), where P_j is j-th Legendre polynomial.
%F From _Paul Barry_, Oct 18 2009: (Start)
%F G.f.: (1-x)/(1-x-2x/(1-x-x/(1-x-x/(1-x-x/(1-... (continued fraction);
%F G.f.: 1/(1-2x/((1-x)^2-x/(1-x/((1-x)^2-x/(1-x/((1-x)^2-x/(1-... (continued fraction);
%F a(n) = Sum_{k = 0..n} (0^(n + k) + C(n + k - 1, 2k - 1)) * C(2k, k) = 0^n + Sum_{k = 0..n} C(n + k - 1, 2k - 1) * C(2k, k). (End)
%F D-finite with recurrence: n*(2*n-3)*a(n) = 2*(6*n^2-12*n+5)*a(n-1) - (n-2)*(2*n-1)*a(n-2). - _Vaclav Kotesovec_, Oct 18 2012
%F a(n) ~ 2^(-1/4)*(3+2*sqrt(2))^n/sqrt(Pi*n). - _Vaclav Kotesovec_, Oct 18 2012
%F a(n) = A277919(2n,n). - _John P. McSorley_, Nov 23 2016
%F a(n) = 2*hypergeom([1 - n, -n], [1], 2) for n>0. - _Peter Luschny_, May 22 2017
%F D-finite with recurrence: n*a(n) +(-7*n+5)*a(n-1) +(7*n-16)*a(n-2) +(-n+3)*a(n-3)=0. - _R. J. Mathar_, Jan 15 2020
%F a(0) = 1; a(n) = (2/n) * Sum_{k=0..n-1} (n^2-k^2) * a(k). - _Seiichi Manyama_, Mar 28 2023
%F G.f.: Sum_{n >= 0} binomial(2*n, n)*x^n/(1 - x)^(2*n) = 1 + 2*x + 10*x^2 + 50*x^3 + .... - _Peter Bala_, Apr 17 2024
%p with(orthopoly): a:=proc(n) if n=0 then 1 else P(n,3)-P(n-1,3) fi end: seq(a(n),n=0..25);
%p a := n -> `if`(n=0, 1, 2*hypergeom([1 - n, -n], [1], 2)):
%p seq(simplify(a(n)), n=0..24); # _Peter Luschny_, May 22 2017
%t CoefficientList[Series[(1 - x)/Sqrt[1 - 6 * x + x^2], {x, 0, 20}], x] (* _Vaclav Kotesovec_, Oct 18 2012 *)
%o (PARI) x='x+O('x^66); Vec((1-x)/sqrt(1-6*x+x^2)) \\ _Joerg Arndt_, May 16 2013
%o (Haskell)
%o a110170 0 = 1
%o a110170 n = a128966 (2 * n) n -- _Reinhard Zumkeller_, Jul 20 2013
%Y Cf. A001850, A110169.
%Y Cf. A085362, A162478, A359758, A360132.
%K nonn,easy
%O 0,2
%A _Emeric Deutsch_, Jul 14 2005
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