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

1, 4, 16, 65, 267, 1105, 4597, 19196, 80380, 337284, 1417582, 5965622, 25130844, 105954110, 447015744, 1886996681, 7969339643, 33670068133, 142301618265, 601586916703, 2543852427847, 10759094481491, 45513214057191, 192560373660245, 814807864164497

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Rob Arthan, Comments on A026674, A026725, A026670
• Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.

Crossrefs

Also a(n) = T(2n-1, n-1), T given by A026670.

Programs

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

Magma

R:=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

Maple

a := n -> add(binomial(2*n,n+k)*combinat:-fibonacci(1+k)*(k/n), k=1..n):
seq(a(n), n=1..30); # Peter Luschny, Apr 28 2016

Mathematica

a[n_] := Sum[Binomial[2n, n+k] Fibonacci[k+1] k/n, {k, 1, n}];
Array[a, 30] (* Jean-François Alcover, Jun 21 2018, after Peter Luschny *)

Maxima

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 */

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

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

Formula

G.f.: (1/2)*((1-x)/(sqrt(1-4*x)-x) - 1). - Ralf Stephan, Feb 05 2004

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

From Gary W. Adamson, Jul 11 2011: (Start)

a(n) is the upper left term in M^n, where M is the following infinite square production matrix:

1, 1, 0, 0, 0, 0, 0, ...

3, 1, 1, 0, 0, 0, 0, ...

6, 1, 1, 1, 0, 0, 0, ...

10, 1, 1, 1, 1, 0, 0, ...

15, 1, 1, 1, 1, 1, 0, ...

21, 1, 1, 1, 1, 1, 1, ...

... (End)

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

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

a(n) ~ (3 - sqrt(5)) * (2 + sqrt(5))^n / (2*sqrt(5)). - Vaclav Kotesovec, Jul 18 2019