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

1,2

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) 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

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 *)

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