cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

A099045 a(n) = (3*0^n + 4^n*binomial(2*n,n))/4.

Original entry on oeis.org

1, 2, 24, 320, 4480, 64512, 946176, 14057472, 210862080, 3186360320, 48432676864, 739699064832, 11342052327424, 174493112729600, 2692179453542400, 41639042214789120, 645405154329231360, 10022762396642181120, 155909637281100595200
Offset: 0

Views

Author

Paul Barry, Sep 24 2004

Keywords

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

Crossrefs

Cf. A069723, A088218, A099044, A099046.

Programs

  • Magma
    [(3*0^n + 4^n*Binomial(2*n, n))/4: n in [ 0..20]]; // Vincenzo Librandi, Nov 24 2012
  • Mathematica
    Join[{1}, Table[4^(n-1)*Binomial[2*n,n], {n,1,30}]] (* G. C. Greubel, Dec 31 2017 *)
  • PARI
    for(n=0,30, print1((3*0^n + 4^n*binomial(2*n,n))/4, ", ")) \\ G. C. Greubel, Dec 31 2017

Formula

G.f.: (1+3*sqrt(1-16*x))/(4*sqrt(1-16*x)).
n*a(n) +8*(-2*n+1)*a(n-1)=0. - R. J. Mathar, Nov 24 2012
E.g.f.: (3 + exp(8*x) * BesselI(0,8*x)) / 4. - Ilya Gutkovskiy, Nov 17 2021

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

Original entry on oeis.org

1, 2, 30, 500, 8750, 157500, 2887500, 53625000, 1005468750, 18992187500, 360851562500, 6888984375000, 132038867187500, 2539208984375000, 48970458984375000, 946762207031250000, 18343517761230468750, 356080050659179687500, 6923778762817382812500
Offset: 0

Views

Author

Paul Barry, Sep 24 2004

Keywords

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

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
Showing 1-2 of 2 results.

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