%I #11 Feb 22 2024 20:22:43

%S 1,2,6,28,206,2418,45970,1440746,75840096,6828414424,1069361760254,

%T 295609883371824,146078092162147126,130419475982163166640,

%U 212257994312591826735888,634463537260289571176650942

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

%H G. C. Greubel, <a href="/A093657/b093657.txt">Table of n, a(n) for n = 1..86</a>

%F a(n) = A093656(2^(n-1)) for n>=1.

%F a(n) = Sum_{k=0..n} A097710(n,k), row sums of triangle A097710.

%t 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[n_]:= A093657[n]= Sum[T[n,k], {k,0,n}];

%t Table[A093657[n], {n,0,30}] (* _G. C. Greubel_, Feb 21 2024 *)

%o (SageMath)

%o @CachedFunction

%o def T(n, k): # T = A097710

%o if n< 0 or k<0 or k>n: return 0

%o elif k==n: return 1

%o elif k==0: return sum(T(n-1,j)*T(j,0) for j in range(n))

%o else: return sum(T(n-1, j)*(T(j, k-1)+T(j,k)) for j in range(n))

%o def A093657(n): return sum(T(n,k) for k in range(n+1))

%o [A093657(n) for n in range(31)] # _G. C. Greubel_, Feb 21 2024

%Y Cf. A093654, A093656.

%Y Related to the number of tournament sequences (A008934).

%Y Cf. A097710, A008934.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Apr 08 2004