A047084 a(n) = Sum_{i=0..n} A047080(i,n-i).

1, 1, 2, 2, 4, 6, 9, 14, 21, 33, 50, 77, 118, 181, 278, 426, 654, 1003, 1539, 2361, 3622, 5557, 8525, 13079, 20065, 30783, 47226, 72452, 111153, 170526, 261614, 401357, 615745, 944650, 1449242, 2223366, 3410994, 5233003, 8028252, 12316605, 18895615, 28988854

Offset: 0

Author

Clark Kimberling

Keywords

• nonn
• easy

Links

• Sean A. Irvine, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A047080, A047081, A047082, A047083, 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-2*j, j): j in [0..Floor(n/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]}];
A047084[n_]:= A047084[n]= Sum[A[2*k-n, n-k], {k,0,n}];
Table[A047084[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-2*j,j) for j in range(1+(n//2))) for n in range(51)] # G. C. Greubel, Oct 31 2022

Formula

a(n) = Sum_{j=0..floor(n/2)} A(n-2*j, j), where A(n,k) = array of A048080(n,k). - G. C. Greubel, Oct 31 2022

Extensions

Entry revised by Sean A. Irvine, May 11 2021