A047074 a(n) = Sum_{i=0..floor(n/2)} T(i,n-i), array T as in A047072.

1, 1, 3, 2, 5, 6, 14, 20, 45, 70, 154, 252, 546, 924, 1980, 3432, 7293, 12870, 27170, 48620, 102102, 184756, 386308, 705432, 1469650, 2704156, 5616324, 10400600, 21544100, 40116600, 82907640, 155117520, 319929885, 601080390, 1237518450

Offset: 0

Author

Clark Kimberling

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A047072, A047073, A047079.

Programs

Magma

b:= func< n | n eq 0 select 1 else 2*Catalan(n-1) >;
function A(n, k)
  if k eq n then return b(n);
  elif k gt n then return Binomial(n+k-1, n) - Binomial(n+k-1, n-1);
  else return Binomial(n+k-1, k) - Binomial(n+k-1, k-1);
  end if; return A;
end function;
[(&+[A(j, n-j): j in [0..Floor(n/2)]]): n in [0..50]]; // G. C. Greubel, Oct 29 2022

Mathematica

A[n_, k_]:= A[n, k]= If[k==n, 2*CatalanNumber[n-1] +2*Boole[n==0], If[k>n, Binomial[n+k-1,n] -Binomial[n+k-1,n-1], Binomial[n+k-1,k] - Binomial[n+k-1, k - 1]]];
A047074[n_]:= Sum[A[j, n-j], {j,0,Floor[n/2]}] +Boole[n==0];
Table[A047074[n], {n, 0, 50}] (* G. C. Greubel, Oct 29 2022 *)

SageMath

def A047072(n, k): # array
    if (k==n): return 2*catalan_number(n-1) + 2*int(n==0)
    elif (k>n): return binomial(n+k-1, n) - binomial(n+k-1, n-1)
    else: return binomial(n+k-1, k) - binomial(n+k-1, k-1)
def A047074(n): return sum( A047072(j, n-j) for j in range((n//2)+1) )
[A047074(n) for n in range(51)] # G. C. Greubel, Oct 29 2022

Extensions

Extra leading 1 removed by Sean A. Irvine, May 11 2021