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A047083 a(n) = Sum_{i=0..floor((n+1)/2)} A047080(n,i).

%I A047083 #17 Nov 01 2022 03:09:10
%S A047083 1,2,2,5,7,15,23,49,76,161,253,532,845,1766,2829,5881,9488,19631,
%T A047083 31863,65649,107112,219857,360360,737152,1213150,2473930,4086217,
%U A047083 8309252,13769519,27927146,46416937,93915759,156520328,315982677,527937429,1063586803,1781131638
%N A047083 a(n) = Sum_{i=0..floor((n+1)/2)} A047080(n,i).
%H A047083 G. C. Greubel, <a href="/A047083/b047083.txt">Table of n, a(n) for n = 0..1000</a>
%t A047083 A[n_, k_]:= Sum[(-1)^j*(n+k-3*j)!/(j!*(n-2*j)!*(k-2*j)!), {j,0,Floor[(n+k)/3]}] -
%t A047083  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 A047083 A047083[n_]:= A047083[n]= Sum[A[n-k,k], {k,0,Floor[(n+1)/2]}];
%t A047083 Table[A047083[n], {n,0,50}] (* _G. C. Greubel_, Oct 31 2022 *)
%o A047083 (Magma)
%o A047083 F:=Factorial;
%o A047083 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 A047083 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 A047083 A:= func< n,k | p(n,k) - q(n,k) >;
%o A047083 [(&+[A(n-j,j): j in [0..Floor((n+1)/2)]]): n in [0..50]]; // _G. C. Greubel_, Oct 31 2022
%o A047083 (SageMath)
%o A047083 f=factorial
%o A047083 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 A047083 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 A047083 def A(n,k): return p(n,k) - q(n,k)
%o A047083 [sum(A(n-j,j) for j in range(1+((n+1)//2))) for n in range(51)] # _G. C. Greubel_, Oct 31 2022
%Y A047083 Cf. A047080, A047081, A047082, A047084, A047085, A047086, A047087, A047088.
%K A047083 nonn
%O A047083 0,2
%A A047083 _Clark Kimberling_
%E A047083 Data corrected by _Sean A. Irvine_, May 11 2021