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A026674 a(n) = T(2n-1,n-1) = T(2n,n+1), T given by A026725.

%I A026674 #46 Jan 12 2024 09:29:23
%S A026674 1,4,16,65,267,1105,4597,19196,80380,337284,1417582,5965622,25130844,
%T A026674 105954110,447015744,1886996681,7969339643,33670068133,142301618265,
%U A026674 601586916703,2543852427847,10759094481491,45513214057191,192560373660245,814807864164497
%N A026674 a(n) = T(2n-1,n-1) = T(2n,n+1), T given by A026725.
%H A026674 G. C. Greubel, <a href="/A026674/b026674.txt">Table of n, a(n) for n = 1..1000</a>
%H A026674 Rob Arthan, <a href="/A026674/a026674.txt">Comments on A026674, A026725, A026670</a>
%H A026674 Paul Barry, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL8/Barry/barry84.html">A Catalan Transform and Related Transformations on Integer Sequences</a>, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
%F A026674 G.f.: (1/2)*((1-x)/(sqrt(1-4*x)-x) - 1). - _Ralf Stephan_, Feb 05 2004
%F A026674 G.f.: x*c(x)^3/(1-x*c(x)^3) = (1-5*x -(1-x)*sqrt(1-4*x))/(2*(x^2+4*x-1)), c(x) the g.f. of A000108. - _Paul Barry_, Mar 19 2007
%F A026674 From _Gary W. Adamson_, Jul 11 2011: (Start)
%F A026674 a(n) is the upper left term in M^n, where M is the following infinite square production matrix:
%F A026674    1, 1, 0, 0, 0, 0, 0, ...
%F A026674    3, 1, 1, 0, 0, 0, 0, ...
%F A026674    6, 1, 1, 1, 0, 0, 0, ...
%F A026674   10, 1, 1, 1, 1, 0, 0, ...
%F A026674   15, 1, 1, 1, 1, 1, 0, ...
%F A026674   21, 1, 1, 1, 1, 1, 1, ...
%F A026674   ... (End)
%F A026674 D-finite with recurrence n*a(n) +(-9*n+8)*a(n-1) +23*(n-2)*a(n-2) +(-11*n+48)*a(n-3) +2*(-2*n+7)*a(n-4)=0. - _R. J. Mathar_, Nov 26 2012
%F A026674 a(n) = (1/n)*Sum_{k=1..n} k*binomial(2*n,n-k)*Sum_{i=0..k/2} binomial(k-i,i). - _Vladimir Kruchinin_, Apr 28 2016
%F A026674 a(n) ~ (3 - sqrt(5)) * (2 + sqrt(5))^n / (2*sqrt(5)). - _Vaclav Kotesovec_, Jul 18 2019
%p A026674 a := n -> add(binomial(2*n,n+k)*combinat:-fibonacci(1+k)*(k/n), k=1..n):
%p A026674 seq(a(n), n=1..30); # _Peter Luschny_, Apr 28 2016
%t A026674 a[n_] := Sum[Binomial[2n, n+k] Fibonacci[k+1] k/n, {k, 1, n}];
%t A026674 Array[a, 30] (* _Jean-François Alcover_, Jun 21 2018, after _Peter Luschny_ *)
%o A026674 (Maxima)
%o A026674 a(n):=sum(k*binomial(2*n,n-k)*(sum(binomial(k-i,i),i,0,k/2)),k,1,n)/n; /* _Vladimir Kruchinin_, Apr 28 2016 */
%o A026674 (PARI) a(n)=sum(k=1,n,k*binomial(2*n,n-k)*sum(i=0,k\2,binomial(k-i,i)))/n \\ _Charles R Greathouse IV_, Apr 28 2016
%o A026674 (Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (-1+5*x +(1-x)*Sqrt(1-4*x))/(2*(1-4*x-x^2)) )); // _G. C. Greubel_, Jul 16 2019
%o A026674 (Sage) a=((-1+5*x +(1-x)*sqrt(1-4*x))/(2*(1-4*x-x^2))).series(x, 30).coefficients(x, sparse=False); a[1:] # _G. C. Greubel_, Jul 16 2019
%o A026674 (GAP) List([1..30], n-> Sum([1..n], k-> Binomial(2*n, n+k)*Fibonacci(k+1) *(k/n) )); # _G. C. Greubel_, Jul 16 2019
%Y A026674 Also a(n) = T(2n-1, n-1), T given by A026670.
%K A026674 nonn
%O A026674 1,2
%A A026674 _Clark Kimberling_