OFFSET
0,3
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..600
FORMULA
a(n) = Sum_{k=0..floor((n+1)/2)} ((k+1)!*C(2*(n-k), n-k)*C(2*k, k)*C(n, 2*k))/((k+1)*(n-k+1)).
a(n) = ( -(n-1)*(n-2)*a(n-1) + 4*(3*n^2 -5*n +1)*a(n-2) + 8*n*(n-2)^2* a(n-3) )/(n+1), with a(0) = 1, a(1) = 1, a(2) = 4. - G. C. Greubel, Jun 25 2024
a(n) ~ 2^(3*n/2 + 1) * exp(sqrt(2*n) - n/2 - 1/2) * n^((n-3)/2) / sqrt(Pi) * (1 - 7/(3*sqrt(2*n))). - Vaclav Kotesovec, Jun 25 2024
MATHEMATICA
a[n_]:= a[n]= If[n<3, (n!)^2, (-(n-1)*(n-2)*a[n-1] +4*(3*n^2-5*n +1)*a[n-2] + 8*(n-2)^2*n*a[n-3])/(n+1)];
Table[a[n], {n, 0, 40}] (* G. C. Greubel, Jun 25 2024 *)
PROG
(PARI) {a(n)=sum(k=0, n\2, binomial(2*n-2*k, n-k)/(n-k+1)*binomial(2*k, k)/(k+1) *(k+1)!*binomial(n, 2*k))}
(Magma)
I:=[1, 1, 4]; [n le 3 select I[n] else (-(n-2)*(n-3)*Self(n-1) + 4*(3*(n-2)^2+n-3)*Self(n-2) + 8*(n-3)^2*(n-1)*Self(n-3))/n: n in [1..30]]; // G. C. Greubel, Jun 25 2024
(SageMath)
@CachedFunction
def a(n): # a = A128321
if n<3: return (1, 1, 4)[n]
else: return (-(n-1)*(n-2)*a(n-1) + 4*(3*n^2-5*n+1)*a(n-2) + 8*(n-2)^2*n*a(n-3))/(n+1)
[a(n) for n in range(31)] # G. C. Greubel, Jun 25 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 25 2007
STATUS
approved