A099045 -id:A099045 - cp's OEIS frontend

A099044 a(n) = (20^n + 3^nbinomial(2*n,n))/3.

1, 2, 18, 180, 1890, 20412, 224532, 2501928, 28146690, 318995820, 3636552348, 41655054168, 479033122932, 5527305264600, 63958818061800, 741922289516880, 8624846615633730, 100454095876204620, 1171964451889053900, 13693479385229998200, 160213708807190978940

Offset: 0

Paul Barry, Sep 24 2004

(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.

Vincenzo Librandi, Table of n, a(n) for n = 0..900

Cf. A069723, A088218, A099045, A099046.

[(2*0^n + 3^n*Binomial(2*n, n))/3: n in [ 0..20]]; // Vincenzo Librandi, Nov 24 2012

Join[{1}, Table[3^(n-1)*binomial(2*n,n), {n,1,30}]] (* G. C. Greubel, Dec 31 2017 *)

for(n=0, 30, print1((2*0^n + 3^n*binomial(2*n,n))/3, ", ")) \\ G. C. Greubel, Dec 31 2017

G.f.: 1/3 + 4*x/(sqrt(1-12*x)(1-sqrt(1-12*x)))  = (1 + 2*sqrt(1-12*x))/(3*sqrt(1-12*x)).
n*a(n) +6*(-2*n+1)*a(n-1)=0. - R. J. Mathar, Nov 24 2012
E.g.f.: (2 + exp(6*x) * BesselI(0,6*x)) / 3. - Ilya Gutkovskiy, Nov 17 2021

A099046 a(n) = (40^n + 5^nbinomial(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

Paul Barry, Sep 24 2004

(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.

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

Cf. A069723, A088218, A099044, A099045.

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

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 *)

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

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

A240558 a(n) = 2^nn!/((floor(n/2)+1)floor(n/2)!^2).

Original entry on oeis.org

1, 2, 4, 24, 32, 320, 320, 4480, 3584, 64512, 43008, 946176, 540672, 14057472, 7028736, 210862080, 93716480, 3186360320, 1274544128, 48432676864, 17611882496, 739699064832, 246566354944, 11342052327424, 3489862254592, 174493112729600, 49855175065600

Offset: 0

Peter Luschny, Apr 14 2014

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A000108, A000984, A056040, A057977, A069723, A099045, A151374, A151403.

A240558 := n -> 2^n*n!/((iquo(n,2)+1)*iquo(n,2)!^2):
seq(A240558(n), n=0..30);

Table[SeriesCoefficient[((I*(2*x*(8*x+1)-1))/Sqrt[16*x^2-1]-2*x+1) /(8*x^2), {x,0,n}], {n,0,22}]

x='x+O('x^50); Vec(round((I*(2*x*(8*x+1)-1))/sqrt(16*x^2-1)-2*x+1) /(8*x^2)) \\ G. C. Greubel, Apr 05 2017

def A240558():
    x, n = 1, 1
    while True:
        yield x
        m = 2*n if is_odd(n) else 8/(n+2)
        x *= m
        n += 1
a = A240558(); [next(a) for i in range(36)]

O.g.f.: ((i*(2*x*(8*x+1)-1))/sqrt(16*x^2-1)-2*x+1) /(8*x^2), where i=sqrt(-1).
For a recurrence see the Sage program.
a(n) = 2^n*A057977(n)
a(2*k) = A151403(k) = 2^k*A151374(k) = 4^k*A000108(k).
a(2*k+1) = A099045(k+1) = 2^k*A069723(k+2) = 4^k*A000984(k+1).
From Peter Luschny, Jan 31 2015: (Start)
a(n) = Sum_{k=0..n} A056040(n)*C(n,k)/(floor(n/2)+1).
a(n) = Sum_{k=0..n} n!*C(n,k)/((floor(n/2)+1)*(floor(n/2)!)^2).
a(n) = 2^n*n!*[x^n]((x+1)*hypergeom([],[2],x^2)).
a(n) ~ 2^(n+N)/((n+1)^*sqrt(Pi*(2*N+1))); here  = 1 if n is even, 0 otherwise and N = n++1. (End)
Conjecture: -(n+2)*(n^2-5)*a(n) +8*(-2*n-1)*a(n-1) +16*(n-1)*(n^2+2*n-4)*a(n-2)=0. - R. J. Mathar, Jun 14 2016

cp's OEIS Frontend

A099044 a(n) = (20^n + 3^nbinomial(2*n,n))/3.

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Magma

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Formula

A099046 a(n) = (40^n + 5^nbinomial(2*n,n))/5.

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Programs

Magma

Mathematica

PARI

Formula

A240558 a(n) = 2^nn!/((floor(n/2)+1)floor(n/2)!^2).

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Programs

Maple

Mathematica

PARI

Sage

Formula

cp's OEIS Frontend

A099044 a(n) = (2*0^n + 3^n*binomial(2*n,n))/3.

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Author

Keywords

Comments

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Programs

Magma

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A240558 a(n) = 2^n*n!/((floor(n/2)+1)*floor(n/2)!^2).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Sage

Formula

A099044 a(n) = (20^n + 3^nbinomial(2*n,n))/3.

A099046 a(n) = (40^n + 5^nbinomial(2*n,n))/5.

A240558 a(n) = 2^nn!/((floor(n/2)+1)floor(n/2)!^2).