login
Expansion of -(1 - 8*x) / sqrt(1 - 4*x).
3

%I #12 Dec 04 2021 06:41:00

%S -1,6,10,28,90,308,1092,3960,14586,54340,204204,772616,2939300,

%T 11232648,43088200,165815280,639859770,2475036900,9593714460,

%U 37255818600,144915581580,564514356120,2201964031800,8599360982160,33619842137700,131570223027048,515366318553912

%N Expansion of -(1 - 8*x) / sqrt(1 - 4*x).

%C Sum_{n>=0} (-a(n)/(-4)^n) is the Cauchy product of Sum_{n>=0} (-A349844(n)/(-8)^n) with itself.

%H Wikipedia, <a href="https://en.m.wikipedia.org/wiki/Cauchy_product">Cauchy product</a>

%F For n > 0, a(n) = 8*binomial(2*(n-1),n-1) - binomial(2*n,n) = binomial(2*(n-1),n-1) * (4 + 2/n).

%F a(n) ~ 4^n * (1/sqrt(Pi*n)).

%e a(1) = binomial(0,0) * (4 + 2/1) = 6;

%e a(2) = binomial(2,1) * (4 + 2/2) = 10;

%e a(3) = binomial(4,2) * (4 + 2/3) = 28;

%e a(4) = binomial(6,3) * (4 + 2/4) = 90.

%o (PARI) a(n) = if(n, binomial(2*(n-1),n-1) * (4 + 2/n), -1)

%Y Cf. A000984, A349844, A349835, A349847.

%K sign,easy

%O 0,2

%A _Jianing Song_, Dec 01 2021