A026726 a(n) = T(2n,n), T given by A026725.
1, 2, 7, 27, 108, 440, 1812, 7514, 31307, 130883, 548547, 2303413, 9686617, 40783083, 171868037, 724837891, 3058850316, 12915186640, 54554594416, 230526280814, 974414815782, 4119854160332, 17422801069670, 73695109608352, 311768697325788, 1319136935150530
Offset: 0
Keywords
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
Programs
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GAP
List([0..30], n-> Sum([0..n], k-> (2*k+1)*Binomial(2*n,n-k)* Fibonacci(k+1)/(n+k+1) )); # G. C. Greubel, Jul 16 2019
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Magma
R
:=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 -
Maple
A026726 := proc(n) A026725(2*n,n) ; end proc: seq(A026726(n),n=0..10) ; # R. J. Mathar, Oct 26 2019
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Mathematica
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 *) -
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 -
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
Formula
From Philippe Deléham, Feb 11 2009: (Start)
From Philippe Deléham, Feb 02 2014: (Start)
a(n) = Sum_{k=0..n} A236843(n+k,2*k).
a(n) = Sum_{k=0..n} A236830(n,k).
a(n) = A236830(n+1,1).
a(n) = A165407(3*n).
G.f.: C(x)/(1-x*C(x)^3), C(x) the g.f. of A000108. (End)
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
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