1,2

G. C. Greubel, Table of n, a(n) for n = 1..1000

a(n+4) = ((4*n^5 + 61*n^4 + 374*n^3 + 1146*n^2 + 1743*n + 1046)*a(n+3) - (2*n^5 + 27*n^4 + 146*n^3 + 380*n^2 + 467*n + 220)*a(n+2) + (n+4)*(n^3 + 10*n^2 + 44*n + 53)*a(n+1) - (n-2)*(n+3)*(n+4)*(n^2 + 8*n + 18)*a(n))/((n+2)*(n+3)*(n+5)*(n^2 + 6*n + 11)). - G. C. Greubel, Oct 30 2022

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]}];

Table[A[n-1, n+1], {n, 50}] (* G. C. Greubel, Oct 30 2022 *)

(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-1, n+1): n in [1..50]]; // G. C. Greubel, Oct 30 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)

[A(n-1, n+1) for n in range(1, 50)] # G. C. Greubel, Oct 30 2022

Cf. A047080, A047081, A047082, A047083, A047084, A047085, A047086, A047088.

Sequence in context: A046342 A238977 A182453 * A000958 A148782 A148783

Adjacent sequences: A047084 A047085 A047086 * A047088 A047089 A047090

Corrected and extended by Sean A. Irvine, May 11 2021

approved