A047083 a(n) = Sum_{i=0..floor((n+1)/2)} A047080(n,i).

1, 2, 2, 5, 7, 15, 23, 49, 76, 161, 253, 532, 845, 1766, 2829, 5881, 9488, 19631, 31863, 65649, 107112, 219857, 360360, 737152, 1213150, 2473930, 4086217, 8309252, 13769519, 27927146, 46416937, 93915759, 156520328, 315982677, 527937429, 1063586803, 1781131638

Offset: 0

Author

Clark Kimberling

Keywords

• nonn

Links

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

Crossrefs

Cf. A047080, A047081, A047082, A047084, A047085, A047086, A047087, A047088.

Programs

Magma

F:=Factorial;
p:= func< n,k | (&+[ (-1)^j*F(n+k-3*j)/(F(j)*F(n-2*j)*F(k-2*j)): j in [0..Min(Floor(n/2), Floor(k/2))]]) >;
q:= func< n,k | n eq 0 or k eq 0 select 0 else (&+[ (-1)^j*F(n+k-3*j-2)/(F(j)*F(n-2*j-1)*F(k-2*j-1)) : j in [0..Min(Floor((n-1)/2), Floor((k-1)/2))]]) >;
A:= func< n,k | p(n,k) - q(n,k) >;
[(&+[A(n-j,j): j in [0..Floor((n+1)/2)]]): n in [0..50]]; // G. C. Greubel, Oct 31 2022

Mathematica

A[n_, k_]:= Sum[(-1)^j*(n+k-3*j)!/(j!*(n-2*j)!*(k-2*j)!), {j,0,Floor[(n+k)/3]}] -
 Sum[(-1)^j*(n+k-3*j-2)!/(j!*(n-2*j-1)!*(k-2*j-1)!), {j,0,Floor[(n+k-2)/3]}];
A047083[n_]:= A047083[n]= Sum[A[n-k,k], {k,0,Floor[(n+1)/2]}];
Table[A047083[n], {n,0,50}] (* G. C. Greubel, Oct 31 2022 *)

SageMath

f=factorial
def p(n,k): return sum( (-1)^j*f(n+k-3*j)/(f(j)*f(n-2*j)*f(k-2*j)) for j in range(1+min((n//2), (k//2))) )
def q(n,k): return sum( (-1)^j*f(n+k-3*j-2)/(f(j)*f(n-2*j-1)*f(k-2*j-1)) for j in range(1+min(((n-1)//2), ((k-1)//2))) )
def A(n,k): return p(n,k) - q(n,k)
[sum(A(n-j,j) for j in range(1+((n+1)//2))) for n in range(51)] # G. C. Greubel, Oct 31 2022

Extensions

Data corrected by Sean A. Irvine, May 11 2021