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A093657 2^(n-1)-th term of the row sums of triangle A093654.

%I A093657 #11 Feb 22 2024 20:22:43
%S A093657 1,2,6,28,206,2418,45970,1440746,75840096,6828414424,1069361760254,
%T A093657 295609883371824,146078092162147126,130419475982163166640,
%U A093657 212257994312591826735888,634463537260289571176650942
%N A093657 2^(n-1)-th term of the row sums of triangle A093654.
%H A093657 G. C. Greubel, <a href="/A093657/b093657.txt">Table of n, a(n) for n = 1..86</a>
%F A093657 a(n) = A093656(2^(n-1)) for n>=1.
%F A093657 a(n) = Sum_{k=0..n} A097710(n,k), row sums of triangle A097710.
%t A093657 T[n_, k_]:= T[n,k]= If[n<0 || k>n, 0, If[n==k, 1, If[k==0, Sum[T[n-1,j]*T[j,0], {j,0,n-1}], Sum[T[n-1,j]*(T[j,k-1]+T[j,k]), {j,0,n-1}] ]]]; (* T = A097710 *)
%t A093657 A093657[n_]:= A093657[n]= Sum[T[n,k], {k,0,n}];
%t A093657 Table[A093657[n], {n,0,30}] (* _G. C. Greubel_, Feb 21 2024 *)
%o A093657 (SageMath)
%o A093657 @CachedFunction
%o A093657 def T(n, k): # T = A097710
%o A093657     if n< 0 or k<0 or k>n: return 0
%o A093657     elif k==n: return 1
%o A093657     elif k==0: return sum(T(n-1,j)*T(j,0) for j in range(n))
%o A093657     else: return sum(T(n-1, j)*(T(j, k-1)+T(j,k)) for j in range(n))
%o A093657 def A093657(n): return sum(T(n,k) for k in range(n+1))
%o A093657 [A093657(n) for n in range(31)] # _G. C. Greubel_, Feb 21 2024
%Y A093657 Cf. A093654, A093656.
%Y A093657 Related to the number of tournament sequences (A008934).
%Y A093657 Cf. A097710, A008934.
%K A093657 nonn
%O A093657 1,2
%A A093657 _Paul D. Hanna_, Apr 08 2004