A070808 - OEIS

%I #7 Aug 04 2022 05:42:23

%S 1,2,42,407,6890,88502,1385727,19762290,303169770,4514031830,

%T 69135179542,1050132147077,16141218975167,247800513084152,

%U 3825796483371170,59118992260132532,916434202205565162

%N Sum(((-1)^k*binomial(4*n,k)),k=n..2*n).

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

%F From _Vaclav Kotesovec_, Aug 04 2022: (Start)

%F Recurrence: 3*(n-1)*n*(2*n - 1)*(3*n - 2)*(3*n - 1)*(43*n^2 - 129*n + 96)*a(n) = 2*(n-1)*(4*n - 3)*(4*n - 1)*(473*n^4 - 1892*n^3 + 2561*n^2 - 1338*n + 216)*a(n-1) + 16*(2*n - 3)*(4*n - 7)*(4*n - 5)*(4*n - 3)*(4*n - 1)*(43*n^2 - 43*n + 10)*a(n-2).

%F a(n) ~ 2^(4*n - 3/2) / sqrt(Pi*n). (End)

%t Table[Sum[(-1)^k Binomial[4n,k],{k,n,2n}],{n,0,20}] (* _Harvey P. Dale_, Nov 20 2014 *)

%K easy,nonn

%O 0,2

%A Sebastian Gutierrez and Sarah Kolitz (skolitz(AT)mit.edu), May 15 2002