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A097713 Column 1 of triangle A097712.

%I A097713 #11 Feb 22 2024 09:09:00
%S A097713 1,3,8,25,111,809,10360,236952,9708797,714862984,95000655195,
%T A097713 22902964060238,10070812803900694,8120691251242651341,
%U A097713 12070960239863869828931,33238610095183531376362138
%N A097713 Column 1 of triangle A097712.
%C A097713 Partial sums of A016121.
%C A097713 The row sums of triangle A097712 give A016121.
%H A097713 G. C. Greubel, <a href="/A097713/b097713.txt">Table of n, a(n) for n = 0..85</a>
%F A097713 a(n) = Sum_{k=0..n} A016121(k).
%t A097713 T[n_, k_]:= T[n, k]= If[n<0 || k>n, 0, If[k==0 || k==n, 1, T[n-1,k] + Sum[T[n-1,j]*T[j,k-1], {j,0,n-1}] ]]; (* T=A097712 *)
%t A097713 A097713[n_]:= T[n,1];
%t A097713 Table[A097713[n], {n,30}] (* _G. C. Greubel_, Feb 22 2024 *)
%o A097713 (SageMath)
%o A097713 @CachedFunction
%o A097713 def T(n, k): # T = A097712
%o A097713     if k<0 or k>n: return 0
%o A097713     elif k==0 or k==n: return 1
%o A097713     else: return T(n-1, k) + sum(T(n-1, j)*T(j, k-1) for j in range(n))
%o A097713 def A097713(n): return T(n,1)
%o A097713 [A097713(n) for n in range(1,31)] # _G. C. Greubel_, Feb 22 2024
%Y A097713 Cf. A016121, A097712.
%K A097713 nonn
%O A097713 0,2
%A A097713 _Paul D. Hanna_, Aug 24 2004