A249513 - OEIS

%I #21 Feb 14 2024 09:18:55

%S 1,5,13,25,51,125,295,625,1345,3125,7173,15625,34269,78125,177153,

%T 390625,864315,1953125,4401655,9765625,21706831,48828125,109676283,

%U 244140625,544031251,1220703125,2736797215,6103515625,13620096675,30517578125,68346531855

%N Expansion of -(4*x*sqrt(4*x^2+1)+8*x^2+1)/((2*x^2-1)*sqrt(4*x^2+1) +4*x^3+x).

%F a(n) = sum(k = 0..n, 4^(n-k)*binomial((n+1)/2,n-k)).

%F a(n) ~ 5^((n+1)/2). - _Vaclav Kotesovec_, Oct 31 2014

%F a(n) = 5^((n+1)/2) if n is odd else a(n) = (4^n*binomial(n/2+1/2, n)* hypergeometric([1, -n], [-n/2+3/2], -1/4)). # _Peter Luschny_, Oct 31 2014

%F D-finite with recurrence: -n*(3*n-4)*a(n) -(n-1)*(3*n-7)*a(n-1) +(3*n^2+8*n+24)*a(n-2) +(3*n^2+2*n+19)*a(n-3) +20*(3*n+2)*(n-3)*a(n-4) +20*(3*n-1)*(n-4)*a(n-5)=0. - _R. J. Mathar_, Jan 25 2020

%t CoefficientList[Series[-(4 x Sqrt[4 x^2 + 1] + 8 x^2 + 1)/((2 x^2 - 1) Sqrt[4 x^2 + 1] + 4 x^3 + x), {x, 0, 40}], x] (* _Vincenzo Librandi_, Oct 31 2014 *)

%t Table[Sum[4^(n-k) Binomial[(n+1)/2,n-k],{k,0,n}],{n,0,30}] (* _Harvey P. Dale_, Jun 10 2023 *)

%o (Maxima)

%o a(n) := sum(4^(n-k)*binomial((n+1)/2,n-k),k,0,n).

%o (Sage)

%o def a(n):

%o     if is_odd(n): return 5^(n//2+1)

%o     return (4^n*binomial(n/2+1/2, n)*hypergeometric([1, -n], [-n/2 +3/2], -1/4)).simplify_hypergeometric()

%o [a(n) for n in range(31)] # _Peter Luschny_, Oct 31 2014

%Y Cf. A000351.

%K nonn

%O 0,2

%A _Vladimir Kruchinin_, Oct 31 2014

%E More terms from _Vincenzo Librandi_, Oct 31 2014