A047079 a(n) = Sum_{i=0..floor(n/2)} A047072(i, n-2*i).

1, 1, 2, 3, 3, 4, 7, 9, 14, 23, 33, 52, 85, 127, 202, 329, 503, 804, 1307, 2027, 3250, 5277, 8263, 13276, 21539, 33957, 54638, 88595, 140373, 226108, 366481, 582865, 939622, 1522487, 2428517, 3917412, 6345929, 10145769, 16374126

Offset: 0

Author

Clark Kimberling

Keywords

• nonn

Links

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

Crossrefs

Cf. A047072, A047073, A047074.

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-2*j): j in [0..Floor(n/2)]]): n in [0..50]]; // G. C. Greubel, Oct 29 2022

Mathematica

T[n_, k_]:= T[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]]];
A047079[n_]:= Sum[T[j, n-2*j], {j,0,Floor[n/2]}] +Boole[n==0];
Table[A047079[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 A047079(n): return sum( A047072(j, n-2*j) for j in range(((n+1)//2)+1) )
[A047079(n) for n in range(51)] # G. C. Greubel, Oct 29 2022

Extensions

Name improved by Sean A. Irvine, May 11 2021