login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A026674 a(n) = T(2n-1,n-1) = T(2n,n+1), T given by A026725. 10
1, 4, 16, 65, 267, 1105, 4597, 19196, 80380, 337284, 1417582, 5965622, 25130844, 105954110, 447015744, 1886996681, 7969339643, 33670068133, 142301618265, 601586916703, 2543852427847, 10759094481491, 45513214057191, 192560373660245 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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.

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) = 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)

Conjecture: 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

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-Fran├žois Alcover, Jun 21 2018, after Peter Luschny *)

PROG

(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

(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

(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

(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

CROSSREFS

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

Sequence in context: A012781 A132820 A165201 * A099781 A026872 A081915

Adjacent sequences:  A026671 A026672 A026673 * A026675 A026676 A026677

KEYWORD

nonn

AUTHOR

Clark Kimberling

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified October 17 04:09 EDT 2019. Contains 328106 sequences. (Running on oeis4.)