A047087 a(n) = A047080(2*n, n+1).

1, 3, 8, 24, 75, 237, 755, 2421, 7804, 25264, 82081, 267487, 873970, 2862038, 9391137, 30869167, 101627704, 335049772, 1106003560, 3655124296, 12092095945, 40042017815, 132712302538, 440207294382, 1461259979347, 4853983051617, 16134233746913, 53660996850207

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

Links

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

Crossrefs

Cf. A047080, A047081, A047082, A047083, A047084, A047085, A047086, 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-1,n+1): n in [1..50]]; // G. C. Greubel, Oct 30 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]}];
Table[A[n-1, n+1], {n, 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

Formula

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

Extensions

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