A097713 - OEIS

%I #11 Feb 22 2024 09:09:00

%S 1,3,8,25,111,809,10360,236952,9708797,714862984,95000655195,

%T 22902964060238,10070812803900694,8120691251242651341,

%U 12070960239863869828931,33238610095183531376362138

%N Column 1 of triangle A097712.

%C Partial sums of A016121.

%C The row sums of triangle A097712 give A016121.

%H G. C. Greubel, <a href="/A097713/b097713.txt">Table of n, a(n) for n = 0..85</a>

%F a(n) = Sum_{k=0..n} A016121(k).

%t 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[n_]:= T[n,1];

%t Table[A097713[n], {n,30}] (* _G. C. Greubel_, Feb 22 2024 *)

%o (SageMath)

%o @CachedFunction

%o def T(n, k): # T = A097712

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

%o     elif k==0 or k==n: return 1

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

%o def A097713(n): return T(n,1)

%o [A097713(n) for n in range(1,31)] # _G. C. Greubel_, Feb 22 2024

%Y Cf. A016121, A097712.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Aug 24 2004