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a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(k,floor(k/2)).
4

%I #18 Feb 08 2017 02:08:25

%S 1,0,2,1,5,5,15,20,50,76,176,286,638,1078,2354,4081,8789,15521,33099,

%T 59279,125477,227239,478193,873885,1830271,3370029,7030571,13027729,

%U 27088871,50469889,104647631,195892564,405187826,761615284,1571990936

%N a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(k,floor(k/2)).

%C Knödel walks starting and ending at 0, with n steps.

%H G. C. Greubel, <a href="/A086905/b086905.txt">Table of n, a(n) for n = 0..1000</a>

%H H. Prodinger, <a href="http://www.mat.univie.ac.at/~slc/wpapers/s50proding.html">The Kernel Method: a collection of examples</a>, Séminaire Lotharingien de Combinatoire, B50f (2004), 19 pp.

%F G.f.: (sqrt((1+2*x)/(1-2*x))-1)/2/x/(1+x).

%F a(n) ~ 2^(n+3/2) / (3*sqrt(Pi*n)) * (1 - 2/(3*n)+ 3*(-1)^n/(4*n)). - _Vaclav Kotesovec_, Mar 02 2014

%t Table[Sum[(-1)^(n-k)*Binomial[k,Floor[k/2]],{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Mar 02 2014 *)

%o (PARI) a(n) = sum(k=0, n, (-1)^(n-k)*binomial(k,k\2)); \\ _Michel Marcus_, Dec 04 2016

%Y Cf. A036256, A001405.

%Y First column of triangle A101491.

%K nonn

%O 0,3

%A _Vladeta Jovovic_, Sep 19 2003