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%I #18 Oct 31 2022 07:33:59
%S 1,3,8,24,75,237,755,2421,7804,25264,82081,267487,873970,2862038,
%T 9391137,30869167,101627704,335049772,1106003560,3655124296,
%U 12092095945,40042017815,132712302538,440207294382,1461259979347,4853983051617,16134233746913,53660996850207
%N a(n) = A047080(2*n, n+1).
%H G. C. Greubel, <a href="/A047087/b047087.txt">Table of n, a(n) for n = 1..1000</a>
%F 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
%t 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]}];
%t Table[A[n-1, n+1], {n, 50}] (* _G. C. Greubel_, Oct 30 2022 *)
%o (Magma)
%o F:=Factorial;
%o 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))]]) >;
%o 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))]]) >;
%o A:= func< n,k | p(n,k) - q(n,k) >;
%o [A(n-1,n+1): n in [1..50]]; // _G. C. Greubel_, Oct 30 2022
%o (SageMath)
%o f=factorial
%o 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))) )
%o 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))) )
%o def A(n,k): return p(n,k) - q(n,k)
%o [A(n-1,n+1) for n in range(1,50)] # _G. C. Greubel_, Oct 30 2022
%Y Cf. A047080, A047081, A047082, A047083, A047084, A047085, A047086, A047088.
%K nonn
%O 1,2
%A _Clark Kimberling_
%E Corrected and extended by _Sean A. Irvine_, May 11 2021
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