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A047084 a(n) = Sum_{i=0..n} A047080(i,n-i).

%I A047084 #18 Nov 01 2022 03:09:15
%S A047084 1,1,2,2,4,6,9,14,21,33,50,77,118,181,278,426,654,1003,1539,2361,3622,
%T A047084 5557,8525,13079,20065,30783,47226,72452,111153,170526,261614,401357,
%U A047084 615745,944650,1449242,2223366,3410994,5233003,8028252,12316605,18895615,28988854
%N A047084 a(n) = Sum_{i=0..n} A047080(i,n-i).
%H A047084 Sean A. Irvine, <a href="/A047084/b047084.txt">Table of n, a(n) for n = 0..1000</a>
%F A047084 a(n) = Sum_{j=0..floor(n/2)} A(n-2*j, j), where A(n,k) = array of A048080(n,k). - _G. C. Greubel_, Oct 31 2022
%t A047084 A[n_, k_]:=Sum[(-1)^j*(n+k-3*j)!/(j!*(n-2*j)!*(k-2*j)!), {j,0,Floor[(n+k)/3]}] -
%t A047084  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 A047084 A047084[n_]:= A047084[n]= Sum[A[2*k-n, n-k], {k,0,n}];
%t A047084 Table[A047084[n], {n, 0, 50}] (* _G. C. Greubel_, Oct 31 2022 *)
%o A047084 (Magma)
%o A047084 F:=Factorial;
%o A047084 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 A047084 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 A047084 A:= func< n,k | p(n,k) - q(n,k) >;
%o A047084 [(&+[A(n-2*j, j): j in [0..Floor(n/2)]]): n in [0..50]]; // _G. C. Greubel_, Oct 31 2022
%o A047084 (SageMath)
%o A047084 f=factorial
%o A047084 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 A047084 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 A047084 def A(n,k): return p(n,k) - q(n,k)
%o A047084 [sum(A(n-2*j,j) for j in range(1+(n//2))) for n in range(51)] # _G. C. Greubel_, Oct 31 2022
%Y A047084 Cf. A047080, A047081, A047082, A047083, A047085, A047086, A047087, A047088.
%K A047084 nonn,easy
%O A047084 0,3
%A A047084 _Clark Kimberling_
%E A047084 Entry revised by _Sean A. Irvine_, May 11 2021