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%I #42 Sep 08 2022 08:45:54
%S 0,1,4,16,68,296,1312,5888,26672,121696,558464,2574848,11917952,
%T 55345408,257741824,1203224576,5629027072,26383656448,123868321792,
%U 582414688256,2742116907008,12926036258816,60998951747584
%N Expansion of o.g.f. (1/2)*(-1 + 1/sqrt(1 - 4*x - 4*x^2)).
%C G.f. A(x) satisfies A(x)^2 + A(x) = (x^2 + x)/(1 - 4*x - 4*x^2). - _Michael Somos_, Jan 28 2019
%H G. C. Greubel, <a href="/A179191/b179191.txt">Table of n, a(n) for n = 0..1000</a>
%H M. Dziemianczuk, <a href="https://arxiv.org/abs/1410.5747">On Directed Lattice Paths With Additional Vertical Steps</a>, arXiv:1410.5747 [math.CO], 2014.
%F G.f.: (1/2)*(-1 + 1/sqrt(1 - 4*x - 4*x^2)).
%F G.f.: A(x) = x*A001006(A000045(x)/x-1)/(1-x*A001006(A000045(x)/x-1)).
%F a(n) = Sum_{m=1..n} m*Sum_{k=m..n} Sum_{i=k..n} binomial(i-1,k-1)* binomial(i,n-i))*Sum_{j=0..k} binomial(j,2*j-m-k)*binomial(k,j))/k)). - _Vladimir Kruchinin_, Mar 11 2011
%F a(n) = Sum_{k=0..n} 2^(n-k-1)*binomial(n,k)*binomial(n-k,k). - _Vladimir Kruchinin_, Mar 12 2015
%F From _Vaclav Kotesovec_, Jan 26 2019: (Start)
%F D-finite with recurrence: n*a(n) = 2*(2*n - 1)*a(n-1) + 4*(n-1)*a(n-2).
%F a(n) ~ 2^(n - 7/4) * (1 + sqrt(2))^(n + 1/2) / sqrt(Pi*n). (End)
%F 0 = a(n)*(16*a(n+1) +24*a(n+2) -8*a(n+3)) + a(n+1)*(+8*a(n+1) +16*a(n+2) -6*a(n+3)) + a(n+2)*(-2*a(n+2) +a(n+3)) for all n in Z except n=-1. - _Michael Somos_, Jan 27 2019
%F a(n) = A006139(n)/2, n>0. - _R. J. Mathar_, Jan 24 2020
%e G.f. = x + 4*x^2 + 16*x^3 + 68*x^4 + 296*x^5 + 1312*x^6 + 5888*x^7 + ....
%t CoefficientList[1/2 (-1 + (1-4x-4x^2)^(-1/2)) + O[x]^23, x] (* _Jean-François Alcover_, Jul 27 2018 *)
%o (Maxima)
%o a(n):=sum(m*sum(sum(binomial(i-1,k-1)*binomial(i,n-i),i,k,n)*sum(binomial(j,2*j-m-k)*binomial(k,j),j,0,k)/k,k,m,n),m,1,n); /* _Vladimir Kruchinin_, Mar 11 2011 */
%o a(n):=sum(2^(n-k-1)*binomial(n,k)*binomial(n-k,k),k,0,n); /* _Vladimir Kruchinin_, Mar 12 2015 */
%o (PARI) my(x='x+O('x^30)); concat([0], Vec((-1 +1/sqrt(1-4*x-4*x^2))/2)) \\ _G. C. Greubel_, Jan 25 2019
%o (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (-1 + 1/Sqrt(1-4*x-4*x^2))/2 )); // _G. C. Greubel_, Jan 25 2019
%o (Sage) ((-1 + 1/sqrt(1-4*x-4*x^2))/2).series(x, 20).coefficients(x, sparse=False) # _G. C. Greubel_, Jan 25 2019
%Y Cf. A178693, A178694, A179190.
%K nonn
%O 0,3
%A _Clark Kimberling_, Jul 01 2010
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