A027173 a(n) = A027170(2n, n-1).

5, 30, 123, 472, 1790, 6794, 25879, 98952, 379674, 1461248, 5638930, 21811540, 84542016, 328287506, 1276868111, 4973645576, 19398954626, 75753497816, 296142817406, 1158865623236, 4539024407576, 17793358627976, 69805770185498, 274055019608372, 1076651019788200

Offset: 1

Author

Clark Kimberling

Keywords

• nonn
• changed

Links

• Stefano Spezia, Table of n, a(n) for n = 1..1600

Crossrefs

Cf. A027170.

Programs

Mathematica

a[n_]:=Binomial[2 n,-1+n]+2 Binomial[2+2 n,n]-4; Array[a,30] (* Stefano Spezia, Sep 02 2025 *)
CoefficientList[Series[(-2 + 2*Sqrt[1-4*x] + 5*x - Sqrt[1-4*x]*x - x^2 + 3*Sqrt[1-4*x]*x^2 - 2*x^3 + 4*Sqrt[1-4*x]*x^3)/(2*Sqrt[1-4*x]*(-1 + x)*x^2),{x,0,25}],x] (* Stefano Spezia, Sep 02 2025 *)
CoefficientList[Series[2 - 4*Exp[x] + 4*Exp[2*x]*BesselI[0, 2*x] + (Exp[2*x]*(5*x - 2)*BesselI[1, 2*x])/x,{x,0,25}],x]*Range[0,25]! (* Stefano Spezia, Sep 02 2025 *)

PARI

my(x='x+O('x^40)); Vec(serlaplace(2 - 4*exp(x) + 4*exp(2*x)*besseli(0, 2*x) + (exp(2*x)*(5*x - 2)*besseli(1, 2*x)))) \\ Michel Marcus, Sep 04 2025

Formula

From Stefano Spezia, Sep 02 2025: (Start)

a(n) = binomial(2*n,n-1) + 2*binomial(2*(1+n),n) - 4.

G.f.: (-2 + 2*sqrt(1-4*x) + 5*x - sqrt(1-4*x)*x - x^2 + 3*sqrt(1-4*x)*x^2 - 2*x^3 + 4*sqrt(1-4*x)*x^3)/(2*sqrt(1-4*x)*(-1 + x)*x^2).

E.g.f.: 2 - 4*exp(x) + 4*exp(2*x)*BesselI(0, 2*x) + (exp(2*x)*(5*x - 2)*BesselI(1, 2*x))/x. (End)

Extensions

a(24)-a(25) from Stefano Spezia, Sep 02 2025