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A047082 a(n) = Sum_{i=0..floor(n/2)} A047080(n,i).

%I A047082 #19 Nov 01 2022 03:09:22
%S A047082 1,1,2,3,7,10,23,34,76,115,253,389,845,1316,2829,4452,9488,15061,
%T A047082 31863,50951,107112,172366,360360,583110,1213150,1972647,4086217,
%U A047082 6673417,13769519,22576008,46416937,76374088,156520328,258371689,527937429,874065163,1781131638
%N A047082 a(n) = Sum_{i=0..floor(n/2)} A047080(n,i).
%H A047082 G. C. Greubel, <a href="/A047082/b047082.txt">Table of n, a(n) for n = 0..1000</a>
%t A047082 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 A047082 A047082[n_]:= A047082[n]= Sum[A[n-k,k], {k,0,Floor[n/2]}];
%t A047082 Table[A047082[n], {n, 0, 50}] (* _G. C. Greubel_, Oct 31 2022 *)
%o A047082 (Magma)
%o A047082 F:=Factorial;
%o A047082 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 A047082 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 A047082 A:= func< n,k | p(n,k) - q(n,k) >;
%o A047082 [(&+[A(n-j,j): j in [0..Floor(n/2)]]): n in [0..50]]; // _G. C. Greubel_, Oct 31 2022
%o A047082 (SageMath)
%o A047082 f=factorial
%o A047082 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 A047082 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 A047082 def A(n,k): return p(n,k) - q(n,k)
%o A047082 [sum(A(n-j,j) for j in range(1+(n//2))) for n in range(51)] # _G. C. Greubel_, Oct 31 2022
%Y A047082 Cf. A047080, A047081, A047083, A047084, A047085, A047086, A047087, A047088.
%K A047082 nonn
%O A047082 0,3
%A A047082 _Clark Kimberling_
%E A047082 Data corrected by _Sean A. Irvine_, May 11 2021