A156361 a(2*n+2) = 7*a(2*n+1), a(2*n+1) = 7*a(2*n) - 6^n*A000108(n), a(0) = 1.

1, 6, 42, 288, 2016, 14040, 98280, 686880, 4808160, 33638976, 235472832, 1647983232, 11535882624, 80745019776, 565215138432, 3956385876480, 27694701135360, 193860506096640, 1357023542676480, 9499115800977408

Offset: 0

Author

Philippe Deléham, Feb 08 2009

Keywords

• nonn

Comments

Hankel transform is 6^C(n+1, 2).

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, A156128, A151162, A156195, A151254, A151281.

Programs

Magma

[n le 3 select Factorial(n+4)/120 else (7*n*Self(n-1) + 24*(n-3)*Self(n-2) - 168*(n-3)*Self(n-3))/n: n in [1..30]]; // G. C. Greubel, Nov 09 2022

Maple

A156361 := proc(n)
    option remember;
    local nh;
    if n= 0 then
        1;
    elif  type(n,'even') then
        7*procname(n-1);
    else
        nh := floor(n/2) ;
        7*procname(n-1)-6^nh*A000108(nh) ;
    end if;
end proc: # R. J. Mathar, Jul 21 2016

Mathematica

a[n_]:= a[n]= If[n==0, 1, 7*a[n-1] -If[EvenQ[n], 0, 6^((n-1)/2)* CatalanNumber[(n-1)/2]]];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Aug 04 2022 *)

SageMath

def a(n): # a = A156361
    if (n==0): return 1
    elif (n%2==1): return 7*a(n-1) - 6^((n-1)/2)*catalan_number((n-1)/2)
    else: return 7*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) * 6^k.

(n+1)*a(n) = 7*(n+1)*a(n-1) + 24*(n-2)*a(n-2) - 168*(n-2)*a(n-3). - R. J. Mathar, Jul 21 2016