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%I #6 Jan 04 2022 20:56:32
%S 1,9,90,891,8910,88938,889380,8890155,88901550,888923646,8889236460,
%T 88889884542,888898845420,8888918303988,88889183039880,
%U 888889778505099,8888897785050990,88888916293698870,888889162936988700
%N a(2*n+2) = 10*a(2*n+1), a(2*n+1) = 10*a(2*n) - 9^n*A000108(n), a(0) = 1.
%C Hankel transform is 9^binomial(n+1,2).
%H G. C. Greubel, <a href="/A156577/b156577.txt">Table of n, a(n) for n = 0..500</a>
%F a(n) = Sum_{k=0..n} A120730(n,k) * 9^k.
%t 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] ]];
%t Table[a[n], {n, 0, 30}] (* _G. C. Greubel_, Jan 04 2022 *)
%o (Sage)
%o def a(n): # a = A156577
%o if (n==0): return 1
%o elif (n%2==1): return 10*a(n-1) - 9^((n-1)/2)*catalan_number((n-1)/2)
%o else: return 10*a(n-1)
%o [a(n) for n in (0..30)] # _G. C. Greubel_, Jan 04 2022
%Y Cf. A000108, A001405, A120730, A151162, A151254, A151281, A156195, A156273, A156361, A156362, A156566.
%K nonn
%O 0,2
%A _Philippe Deléham_, Feb 10 2009
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