A299845 - OEIS

%I #17 May 23 2024 04:39:13

%S 1,1,25,425,7025,116625,1951625,32903225,558265825,9522632225,

%T 163160773625,2806202183625,48420275891025,837813745045425,

%U 14531896733426025,252593595973313625,4398859688478578625,76733590756134492225,1340547988367851940825,23451231922182584693225

%N a(n) = hypergeom([-n, n - 1], [1], -4).

%H Robert Israel, <a href="/A299845/b299845.txt">Table of n, a(n) for n = 0..798</a>

%F 4*n*(n-2)^2*a(n) + 4*(n-1)^2*(n-3)*a(n-2) - 4*(2*n-3)*(9*n^2-27*n+17)*a(n-1) = 0.  - _Robert Israel_, Mar 21 2018

%F a(n) ~ 2^(-3/2) * 5^(3/4) * phi^(6*n - 3) / sqrt(Pi*n), where phi = A001622 = (1 + sqrt(5))/2 is the golden ratio. - _Vaclav Kotesovec_, Jul 05 2018

%F a(n) = 4^n*Sum_{k=0..n} (5/4)^k*(Gamma(n + 1)*Gamma(n - 1)/(Gamma(k + 1)*Gamma(n - k + 1)^2*Gamma(k - 1)) for n >= 2. - _Detlef Meya_, May 22 2024

%p f:= gfun:-rectoproc({4*n*(n-2)^2*a(n)+4*(n-1)^2*(n-3)*a(n-2)-4*(2*n-3)*(9*n^2-27*n+17)*a(n-1)=0,

%p a(0)=1,a(1)=1,a(2)=25},a(n),remember):

%p map(f, [$0..100]); # _Robert Israel_, Mar 21 2018

%t a[n_] := Hypergeometric2F1[-n, n - 1, 1, -4]; Table[a[n], {n, 0, 19}]

%t a[0]:=1; a[1]:=1; a[n_] := 4^n*Sum[(5/4)^k*(Gamma[n + 1]*Gamma[n - 1])/(Gamma[k + 1]*Gamma[n - k + 1]^2*Gamma[k - 1]),{k,0,n}]; Flatten[Table[a[n],{n,0,19}]] (* _Detlef Meya_, May 22 2024 *)

%Y Cf. A299506, A243946, A084769, A243947.

%Y Cf. A300945, A300946.

%K nonn

%O 0,3

%A _Peter Luschny_, Mar 16 2018