%I #21 Mar 28 2023 14:00:45
%S 1,2,14,88,566,3722,24856,167868,1143462,7841434,54065574,374437404,
%T 2602879712,18150990238,126918338116,889551010728,6247598686710,
%U 43958881741086,309801915943318,2186512103767096,15452093394793006
%N Expansion of 1/sqrt(1-4x/(1-x)^4).
%C Partial sums are A162479.
%H Robert Israel, <a href="/A162478/b162478.txt">Table of n, a(n) for n = 0..1163</a>
%F G.f.: 1/(1-2x/((1-x)^4-x/(1-x/((1-x)^4-x/(1-... (continued fraction);
%F a(n) = Sum_{k=0..n} C(n+3k-1,n-k)*A000984(k).
%F D-finite with recurrence: n*a(n) +(7-9n)*a(n-1) +2*(7n-17)*a(n-2) +10*(3-n)*a(n-3) +5*(n-4)*a(n-4) +(5-n)*a(n-5)=0. - _R. J. Mathar_, Nov 17 2011
%F Recurrence confirmed using the differential equation (6*x+2)*g+(x^5-5*x^4+10*x^3-14*x^2+9*x-1)*g'=0 satisfied by the G.f. - _Robert Israel_, Dec 27 2017
%F a(0) = 1; a(n) = (2/n) * Sum_{k=0..n-1} (n+k) * binomial(n+2-k,3) * a(k). - _Seiichi Manyama_, Mar 28 2023
%p f:= gfun:-rectoproc({n*a(n) +(7-9*n)*a(n-1) +2*(7*n-17)*a(n-2) +10*(3-n)*a(n-3) +5*(n-4)*a(n-4) +(5-n)*a(n-5),a(0)=1,a(1)=2,a(2)=14,a(3)=88,a(4)=566},a(n),remember):
%p map(f, [$0..50]); # _Robert Israel_, Dec 27 2017
%t CoefficientList[Series[1/Sqrt[1-(4x)/(1-x)^4],{x,0,20}],x] (* _Harvey P. Dale_, Aug 02 2016 *)
%Y Cf. A000984, A162479.
%K easy,nonn
%O 0,2
%A _Paul Barry_, Jul 04 2009