A305032 a(0) = 0, a(1) = 1 and a(n) = 6*a(n-1)/(n-1) + 4*a(n-2) for n > 1.

0, 1, 6, 22, 68, 190, 500, 1260, 3080, 7350, 17220, 39732, 90552, 204204, 456456, 1012440, 2230800, 4886310, 10647780, 23094500, 49884120, 107343236, 230205976, 492156392, 1049212528, 2230928700, 4732273000, 10015777800, 21154820400, 44596287000, 93846099600

Offset: 0

Author

Seiichi Manyama, May 24 2018

Keywords

• nonn

Comments

Let a(0) = 0, a(1) = 1 and a(n) = 2*m*a(n-1)/(n-1) + k^2*a(n-2), for n > 1, then the g.f. is x/(2*m) * d/dx ((1 + k*x)/(1 - k*x))^(m/k).

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..3000

Crossrefs

Cf. A100071, A304944, A305031.

Programs

Magma

[n le 2 select n-1 else 2*(3*Self(n-1) + 2*(n-2)*Self(n-2))/(n-2): n in [1..40]]; // G. C. Greubel, Jun 07 2023

Mathematica

CoefficientList[Series[x*Sqrt[1-4*x^2]/(1-2*x)^3, {x,0,40}], x] (* G. C. Greubel, Jun 07 2023 *)

SageMath

@CachedFunction
def a(n): # b = A305032
    if n<2: return n
    else: return 2*(3*a(n-1) + 2*(n-1)*a(n-2))//(n-1)
[a(n) for n in range(41)] # G. C. Greubel, Jun 07 2023

Formula

a(n) = n*A305031(n)/6.

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