A156577 a(2*n+2) = 10*a(2*n+1), a(2*n+1) = 10*a(2*n) - 9^n*A000108(n), a(0) = 1.

1, 9, 90, 891, 8910, 88938, 889380, 8890155, 88901550, 888923646, 8889236460, 88889884542, 888898845420, 8888918303988, 88889183039880, 888889778505099, 8888897785050990, 88888916293698870, 888889162936988700

Offset: 0

Author

Philippe Deléham, Feb 10 2009

Keywords

• nonn

Comments

Hankel transform is 9^binomial(n+1,2).

Links

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

Crossrefs

Cf. A000108, A001405, A120730, A151162, A151254, A151281, A156195, A156273, A156361, A156362, A156566.

Programs

Mathematica

a[n_]:= a[n]= If[n==0, 1, If[OddQ[n], 10*a[n-1] -9^((n-1)/2)*CatalanNumber[(n-1)/2], 10*a[n-1] ]];
Table[a[n], {n, 0, 30}] (* G. C. Greubel, Jan 04 2022 *)

Sage

def a(n): # a = A156577
    if (n==0): return 1
    elif (n%2==1): return 10*a(n-1) - 9^((n-1)/2)*catalan_number((n-1)/2)
    else: return 10*a(n-1)
[a(n) for n in (0..30)] # G. C. Greubel, Jan 04 2022

Formula

a(n) = Sum_{k=0..n} A120730(n,k) * 9^k.