A047088 a(n) = A047080(2*n+1, n+2).

1, 4, 12, 37, 118, 380, 1229, 3989, 12987, 42394, 138709, 454768, 1493690, 4913969, 16189534, 53407853, 176397299, 583242159, 1930349545, 6394665589, 21201345460, 70346920007, 233581374587, 776105485336, 2580316142887, 8583746045611, 28570407158100

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, A047087.

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+2): n in [1..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]}];
Table[A[n-1, n+2], {n, 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)
[A(n-1,n+2) for n in range(1,50)] # G. C. Greubel, Oct 31 2022

Formula

a(n+4) = ((16*n^5 + 324*n^4 + 2624*n^3 + 10509*n^2 + 20655*n + 15930)*a(n+3) - (8*n^5 + 148*n^4 + 1090*n^3 + 3953*n^2 + 7365*n + 5994)*a(n+2) + (4*n^4 + 84*n^3 + 701*n^2 + 2451*n + 2646)*a(n+1) - (n-3)*(n+6)*(2*n+7)*(2*n^2 + 23*n + 72)*a(n) )/((n+3)*(n+6)*(2*n+5)*(2*n^2 + 19*n + 51)). - G. C. Greubel, Oct 31 2022

Extensions

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