A099046 a(n) = (4*0^n + 5^n*binomial(2*n,n))/5.

1, 2, 30, 500, 8750, 157500, 2887500, 53625000, 1005468750, 18992187500, 360851562500, 6888984375000, 132038867187500, 2539208984375000, 48970458984375000, 946762207031250000, 18343517761230468750, 356080050659179687500, 6923778762817382812500

Offset: 0

Author

Paul Barry, Sep 24 2004

Keywords

• easy
• nonn

Comments

(1 + (k-1)*sqrt(1-4*k*x))/(k*sqrt(1-4*k*x)) is the g.f. for ((k-1)*0^n + k^n*binomial(2*n,n))/k.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..760

Crossrefs

Cf. A069723, A088218, A099044, A099045.

Programs

Magma

[(4*0^n + 5^n*Binomial(2*n, n))/5: n in [ 0..30]]; // G. C. Greubel, Dec 31 2017

Mathematica

CoefficientList[Series[(1+4Sqrt[1-20x])/(5Sqrt[1-20x]),{x,0,20}],x]  (* Harvey P. Dale, Mar 30 2011 *)
Join[{1}, Table[5^(n - 1)*Binomial[2*n, n], {n,1,50}]] (* G. C. Greubel, Dec 31 2017 *)

PARI

for(n=0,30, print1((4*0^n + 5^n*binomial(2*n,n))/5, ", ")) \\ G. C. Greubel, Dec 31 2017

Formula

G.f.: (1 + 4*sqrt(1-20*x))/(5*sqrt(1-20*x)).

n*a(n) +10*(-2*n+1)*a(n-1)=0. - R. J. Mathar, Nov 24 2012

E.g.f.: (4 + exp(10*x) * BesselI(0,10*x)) / 5. - Ilya Gutkovskiy, Nov 17 2021

a(n) = Integral_{x = 0..20} x^n * w(x) dx for n >= 1, where w(x) = 1/( 5*Pi*sqrt(x*(20 - x)) ) is positive on the interval (0, 20). The weight function w(x) is singular at x = 0 and at x = 20 and is the solution of the Hausdorff moment problem. - Peter Bala, Oct 12 2024