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A118928 a(n) = Sum_{k=0..floor(n/2)} C(n-k,k)*C(n-k,k+1)/(n-k) * a(k), with a(0)=1.

%I A118928 #9 Nov 24 2021 20:13:20
%S A118928 1,1,1,2,4,8,17,38,92,238,643,1790,5076,14573,42241,123484,364052,
%T A118928 1082602,3247759,9829820,30019326,92517644,287805801,903822922,
%U A118928 2865339252,9168572009,29601077285,96377791839,316264456921
%N A118928 a(n) = Sum_{k=0..floor(n/2)} C(n-k,k)*C(n-k,k+1)/(n-k) * a(k), with a(0)=1.
%C A118928 Invariant column vector V under matrix product A089732 *V = V: a(n) = Sum_{k=0,[n/2]} A089732 (n,k)*a(k), where A089732(n,k) = number of peakless Motzkin paths of length n having k (1,1) steps.
%H A118928 G. C. Greubel, <a href="/A118928/b118928.txt">Table of n, a(n) for n = 0..1000</a>
%F A118928 a(n) = Sum_{k=0..floor(n/2)} C(n-k,k)*C(n-k,k+1)/(n-k) * a(k), with a(0)=1.
%t A118928 a[n_]:= a[n]= If[n==0, 1, Sum[Binomial[n-k, k]*Binomial[n-k, k+1]*a[k]/(n-k), {k, 0, Floor[n/2]}]];
%t A118928 Table[a[n], {n, 0, 30}] (* _G. C. Greubel_, Nov 24 2021 *)
%o A118928 (PARI) {a(n)=if(n==0,1,sum(k=0,n\2,binomial(n-k,k)*binomial(n-k,k+1)/(n-k)*a(k)))}
%o A118928 (Sage)
%o A118928 @CachedFunction
%o A118928 def A118928(n):
%o A118928     if (n==0): return 1
%o A118928     else: return sum( binomial(n-k, k)*binomial(n-k, k+1)*A118928(k)/(n-k) for k in (0..n//2) )
%o A118928 [A118928(n) for n in (0..30)] # _G. C. Greubel_, Nov 24 2021
%Y A118928 Cf. A089732.
%K A118928 nonn
%O A118928 0,4
%A A118928 _Paul D. Hanna_, May 06 2006