A097713 Column 1 of triangle A097712.

1, 3, 8, 25, 111, 809, 10360, 236952, 9708797, 714862984, 95000655195, 22902964060238, 10070812803900694, 8120691251242651341, 12070960239863869828931, 33238610095183531376362138

Offset: 0

Author

Paul D. Hanna, Aug 24 2004

Keywords

• nonn

Comments

Partial sums of A016121.

The row sums of triangle A097712 give A016121.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..85

Crossrefs

Cf. A016121, A097712.

Programs

Mathematica

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 *)
A097713[n_]:= T[n,1];
Table[A097713[n], {n,30}] (* G. C. Greubel, Feb 22 2024 *)

SageMath

@CachedFunction
def T(n, k): # T = A097712
    if k<0 or k>n: return 0
    elif k==0 or k==n: return 1
    else: return T(n-1, k) + sum(T(n-1, j)*T(j, k-1) for j in range(n))
def A097713(n): return T(n,1)
[A097713(n) for n in range(1,31)] # G. C. Greubel, Feb 22 2024

Formula

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