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A047086 a(n) = T(2*n+1, n), array T as in A047080.

%I A047086 #18 Oct 31 2022 07:34:03
%S A047086 1,2,5,15,46,143,450,1429,4570,14698,47491,154042,501283,1635835,
%T A047086 5351138,17541671,57610988,189521640,624389105,2059824523,6803433916,
%U A047086 22495796651,74457478476,246667937610,817866796549,2713874203112,9011747680649,29944572743724
%N A047086 a(n) = T(2*n+1, n), array T as in A047080.
%H A047086 G. C. Greubel, <a href="/A047086/b047086.txt">Table of n, a(n) for n = 0..1000</a>
%F A047086 a(n+4) = ((16*n^3 + 100*n^2 + 188*n + 105)*a(n+3) - (8*n^3 + 36*n^2 + 46*n + 5)*a(n+2) + (4*n^2 + 16*n + 25)*a(n+1) - (n-1)*(2*n+5)^2*a(n))/((n+4)*(2*n+3)^2). - _G. C. Greubel_, Oct 30 2022
%t A047086 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 A047086 T[n_, k_]:= A[n-k,k];
%t A047086 Table[T[2*n+1,n], {n,0,50}] (* _G. C. Greubel_, Oct 30 2022 *)
%o A047086 (Magma)
%o A047086 F:=Factorial;
%o A047086 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 A047086 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 A047086 A:= func< n,k | p(n,k) - q(n,k) >;
%o A047086 [A(n+1,n): n in [0..50]]; // _G. C. Greubel_, Oct 30 2022
%o A047086 (SageMath)
%o A047086 f=factorial
%o A047086 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 A047086 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 A047086 def A(n,k): return p(n,k) - q(n,k)
%o A047086 [A(n+1,n) for n in range(51)] # _G. C. Greubel_, Oct 30 2022
%Y A047086 Cf. A047080, A047081, A047082, A047083, A047084, A047085, A047087, A047088.
%K A047086 nonn
%O A047086 0,2
%A A047086 _Clark Kimberling_
%E A047086 Corrected and extended by _Sean A. Irvine_, May 11 2021