A047084 - OEIS

%I #18 Nov 01 2022 03:09:15

%S 1,1,2,2,4,6,9,14,21,33,50,77,118,181,278,426,654,1003,1539,2361,3622,

%T 5557,8525,13079,20065,30783,47226,72452,111153,170526,261614,401357,

%U 615745,944650,1449242,2223366,3410994,5233003,8028252,12316605,18895615,28988854

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

%H Sean A. Irvine, <a href="/A047084/b047084.txt">Table of n, a(n) for n = 0..1000</a>

%F 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 A[n_, k_]:=Sum[(-1)^j*(n+k-3*j)!/(j!*(n-2*j)!*(k-2*j)!), {j,0,Floor[(n+k)/3]}] -

%t  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[n_]:= A047084[n]= Sum[A[2*k-n, n-k], {k,0,n}];

%t Table[A047084[n], {n, 0, 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-2*j, j): j in [0..Floor(n/2)]]): n in [0..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 [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 Cf. A047080, A047081, A047082, A047083, A047085, A047086, A047087, A047088.

%K nonn,easy

%O 0,3

%A _Clark Kimberling_

%E Entry revised by _Sean A. Irvine_, May 11 2021