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%I #16 Oct 31 2022 07:34:09
%S 1,4,12,37,118,380,1229,3989,12987,42394,138709,454768,1493690,
%T 4913969,16189534,53407853,176397299,583242159,1930349545,6394665589,
%U 21201345460,70346920007,233581374587,776105485336,2580316142887,8583746045611,28570407158100
%N a(n) = A047080(2*n+1, n+2).
%H G. C. Greubel, <a href="/A047088/b047088.txt">Table of n, a(n) for n = 1..1000</a>
%F 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
%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+2], {n, 50}] (* _G. C. Greubel_, Oct 31 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+2): n in [1..50]]; // _G. C. Greubel_, Oct 31 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+2) for n in range(1,50)] # _G. C. Greubel_, Oct 31 2022
%Y Cf. A047080, A047081, A047082, A047083, A047084, A047085, A047086, A047087.
%K nonn
%O 1,2
%A _Clark Kimberling_
%E Corrected and extended by _Sean A. Irvine_, May 11 2021
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