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%I #16 Jan 30 2020 21:29:15
%S 1,8,104,1472,21856,333568,5183744,81590272,1296426496,20750839808,
%T 334081306624,5404163080192,87763693060096,1430025994108928,
%U 23367175920287744,382767375745810432,6283401962864377856
%N Expansion of 1/sqrt(1-16x-16x^2).
%C Central coefficient of (1+8x+20x^2)^n. Eighth binomial transform of 1/sqrt(1-80x^2). In general, 1/sqrt(1-4*r*x-4*r*x^2) has e.g.f. exp(2rx)BesselI(0,2r*sqrt((r+1)/r)x)), a(n)=sum{k=0..n, C(2k,k)C(k,n-k)r^k}, gives the central coefficient of (1+(2r)x+r(r+1)x^2) and is the (2r)-th binomial transform of 1/sqrt(1-8*C(n+1,2)x^2).
%H Vincenzo Librandi, <a href="/A106260/b106260.txt">Table of n, a(n) for n = 0..200</a>
%H Hacène Belbachir, Abdelghani Mehdaoui, László Szalay, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL22/Szalay/szalay42.html">Diagonal Sums in the Pascal Pyramid, II: Applications</a>, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.
%F E.g.f.: exp(8*x)*BesselI(0, 8*sqrt(5/4)*x); a(n)=sum{k=0..n, C(2k, k)C(k, n-k)4^k}.
%F D-finite with recurrence: n*a(n) = 8*(2*n-1)*a(n-1) + 16*(n-1)*a(n-2). - _Vaclav Kotesovec_, Oct 17 2012
%F a(n) ~ sqrt(50+20*sqrt(5))*(8+4*sqrt(5))^n/(10*sqrt(Pi*n)). - _Vaclav Kotesovec_, Oct 17 2012
%t CoefficientList[Series[1/Sqrt[1-16*x-16*x^2], {x, 0, 20}], x] (* _Vaclav Kotesovec_, Oct 17 2012 *)
%Y Cf. A006139, A106258, A106259, A106261.
%K easy,nonn
%O 0,2
%A _Paul Barry_, Apr 28 2005
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