A156566 a(2n+2) = 9*a(2n+1), a(2n+1) = 9*a(2n) - 8^n*A000108(n), a(0)=1.

1, 8, 72, 640, 5760, 51712, 465408, 4186112, 37675008, 339017728, 3051159552, 27459059712, 247131537408, 2224149233664, 20017343102976, 180155188248576, 1621396694237184, 14592546256715776, 131332916310441984

Offset: 0

Author

Philippe Deléham, Feb 10 2009

Keywords

• nonn

Comments

Hankel transform is 8^C(n+1,2).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..500

Crossrefs

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

Cf. A156195, A156270, A156361, A156362, A156577.

Cf. A001018, A120730.

Programs

Mathematica

a[0] = 1; a[1] = 8; a[2] = 72; a[n_] := a[n] = (-288*(n-2)*a[n-3] + 32*(n-2)*a[n-2] + 9*(n+1)*a[n-1])/(n+1); Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Nov 15 2016 *)
a[n_]:= a[n]= If[n==0, 1, If[OddQ[n], 9*a[n-1] - 8^((n-1)/2)*CatalanNumber[(n- 1)/2], 9*a[n-1]]]; Table[a[n], {n,0,30}] (* G. C. Greubel, May 18 2022 *)

SageMath

def a(n): # a = A156566
    if (n==0): return 1
    elif (n%2==1): return 9*a(n-1) - 8^((n-1)/2)*catalan_number((n-1)/2)
    else: return 9*a(n-1)
[a(n) for n in (0..30)] # G. C. Greubel, May 18 2022

Formula

a(n) = Sum_{k=0..n} A120730(n,k)*8^k.