A026726 - OEIS

%I #26 Sep 15 2022 20:53:36

%S 1,2,7,27,108,440,1812,7514,31307,130883,548547,2303413,9686617,

%T 40783083,171868037,724837891,3058850316,12915186640,54554594416,

%U 230526280814,974414815782,4119854160332,17422801069670,73695109608352,311768697325788,1319136935150530

%N a(n) = T(2n,n), T given by A026725.

%H G. C. Greubel, <a href="/A026726/b026726.txt">Table of n, a(n) for n = 0..1000</a>

%H 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 From _Philippe Deléham_, Feb 11 2009: (Start)

%F a(n) = Sum_{k=0..n} A039599(n,k)*A000045(k+1).

%F a(n) = Sum_{k=0..n} A106566(n,k)*A122367(k). (End)

%F From _Philippe Deléham_, Feb 02 2014: (Start)

%F a(n) = Sum_{k=0..n} A236843(n+k,2*k).

%F a(n) = Sum_{k=0..n} A236830(n,k).

%F a(n) = A236830(n+1,1).

%F a(n) = A165407(3*n).

%F G.f.: C(x)/(1-x*C(x)^3), C(x) the g.f. of A000108. (End)

%F n*(5*n-11)*a(n) +2*(-20*n^2+59*n-30)*a(n-1) +15*(5*n^2-19*n+16)*a(n-2) +2*(5*n-6)*(2*n-5)*a(n-3)=0. - _R. J. Mathar_, Oct 26 2019

%F n*a(n) +(-7*n+4)*a(n-1) +(7*n-2)*a(n-2) +(19*n-60)*a(n-3) +2*(2*n-7)*a(n-4)=0. - _R. J. Mathar_, Oct 26 2019

%p A026726 := proc(n)

%p     A026725(2*n,n) ;

%p end proc:

%p seq(A026726(n),n=0..10) ; # _R. J. Mathar_, Oct 26 2019

%t CoefficientList[Series[4*x*(1-Sqrt[1-4*x])/(8*x^2-(1-Sqrt[1-4*x])^3), {x,0,30}], x] (* _G. C. Greubel_, Jul 16 2019 *)

%o (PARI) my(x='x+O('x^30)); Vec(4*x*(1-sqrt(1-4*x))/(8*x^2-(1-sqrt(1-4*x))^3)) \\ _G. C. Greubel_, Jul 16 2019

%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 4*x*(1-Sqrt(1-4*x))/(8*x^2-(1-Sqrt(1-4*x))^3) )); // _G. C. Greubel_, Jul 16 2019

%o (Sage) (4*x*(1-sqrt(1-4*x))/(8*x^2-(1-sqrt(1-4*x))^3)).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, Jul 16 2019

%o (GAP) List([0..30], n-> Sum([0..n], k-> (2*k+1)*Binomial(2*n,n-k)*

%o Fibonacci(k+1)/(n+k+1) )); # _G. C. Greubel_, Jul 16 2019

%Y Cf. A000045, A000108, A026725.

%K nonn

%O 0,2

%A _Clark Kimberling_