A047086 - OEIS

%I #18 Oct 31 2022 07:34:03

%S 1,2,5,15,46,143,450,1429,4570,14698,47491,154042,501283,1635835,

%T 5351138,17541671,57610988,189521640,624389105,2059824523,6803433916,

%U 22495796651,74457478476,246667937610,817866796549,2713874203112,9011747680649,29944572743724

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

%H G. C. Greubel, <a href="/A047086/b047086.txt">Table of n, a(n) for n = 0..1000</a>

%F 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 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 T[n_, k_]:= A[n-k,k];

%t Table[T[2*n+1,n], {n,0,50}] (* _G. C. Greubel_, Oct 30 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+1,n): n in [0..50]]; // _G. C. Greubel_, Oct 30 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 [A(n+1,n) for n in range(51)] # _G. C. Greubel_, Oct 30 2022

%Y Cf. A047080, A047081, A047082, A047083, A047084, A047085, A047087, A047088.

%K nonn

%O 0,2

%A _Clark Kimberling_

%E Corrected and extended by _Sean A. Irvine_, May 11 2021