A047086 a(n) = T(2*n+1, n), array T as in A047080.

1, 2, 5, 15, 46, 143, 450, 1429, 4570, 14698, 47491, 154042, 501283, 1635835, 5351138, 17541671, 57610988, 189521640, 624389105, 2059824523, 6803433916, 22495796651, 74457478476, 246667937610, 817866796549, 2713874203112, 9011747680649, 29944572743724

Offset: 0

Author

Clark Kimberling

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

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

Programs

Magma

F:=Factorial;
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))]]) >;
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))]]) >;
A:= func< n,k | p(n,k) - q(n,k) >;
[A(n+1,n): n in [0..50]]; // G. C. Greubel, Oct 30 2022

Mathematica

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[n_, k_]:= A[n-k,k];
Table[T[2*n+1,n], {n,0,50}] (* G. C. Greubel, Oct 30 2022 *)

SageMath

f=factorial
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))) )
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))) )
def A(n,k): return p(n,k) - q(n,k)
[A(n+1,n) for n in range(51)] # G. C. Greubel, Oct 30 2022

Formula

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

Extensions

Corrected and extended by Sean A. Irvine, May 11 2021