A156195 a(2n+2) = 6*a(2n+1), a(2n+1) = 6*a(2n) - 5^n*A000108(n), a(0)=1.

1, 5, 30, 175, 1050, 6250, 37500, 224375, 1346250, 8068750, 48412500, 290343750, 1742062500, 10450312500, 62701875000, 376177734375, 2257066406250, 13541839843750, 81251039062500, 487496738281250, 2924980429687500, 17549718554687500, 105298311328125000

Offset: 0

Author

Philippe Deléham, Feb 05 2009

Keywords

• nonn

Comments

Hankel transform is 5^C(n+1,2). - Philippe Deléham, Feb 05 2009

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Paul Barry, A Note on a One-Parameter Family of Catalan-Like Numbers, JIS 12 (2009) 09.5.4.

Crossrefs

Cf. A000108, A001405, A120730, A151162, A151254, A151281, A156058.

Programs

Magma

[n le 3 select Factorial(n+3)/24 else (6*n*Self(n-1) + 20*(n-3)*Self(n-2) - 120*(n-3)*Self(n-3))/n: n in [1..30]]; // G. C. Greubel, Nov 09 2022

Maple

A156195 := proc(n)
    option remember;
    local nh;
    if n= 0 then
        1;
    elif  type(n,'even') then
        6*procname(n-1);
    else
        nh := floor(n/2) ;
        6*procname(n-1)-5^nh*A000108(nh) ;
    end if;
end proc: # R. J. Mathar, Jul 21 2016

Mathematica

CoefficientList[Series[(Sqrt[1-20x^2]+10x-1)/(10x(1-6x)),{x,0,30}],x] (* Harvey P. Dale, Oct 21 2016 *)

SageMath

def a(n): # a = A156195
    if (n==0): return 1
    elif (n%2==1): return 6*a(n-1) - 5^((n-1)/2)*catalan_number((n-1)/2)
    else: return 6*a(n-1)
[a(n) for n in (0..30)] # G. C. Greubel, Nov 09 2022

Formula

a(n) = Sum_{k=0..n} A120730(n,k)*5^k.

G.f.: (sqrt(1-20*x^2) +10*x -1)/(10*x*(1-6*x)). - Philippe Deléham, Feb 05 2009

(n+1)*a(n) = 6*(n+1)*a(n-1) + 20*(n-2)*a(n-2) - 120*(n-2)*a(n-3). - R. J. Mathar, Jul 21 2016

Extensions

Corrected and extended by Harvey P. Dale, Oct 21 2016