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%I A106258 #25 Jan 30 2020 21:29:15
%S A106258 1,4,28,208,1624,13024,106336,879232,7338592,61699456,521753728,
%T A106258 4433024512,37812715264,323603221504,2777262164992,23893731463168,
%U A106258 206005885076992,1779480850438144,15396895523989504,133420304211238912
%N A106258 Expansion of 1/sqrt(1-8x-8x^2).
%C A106258 Central coefficient of (1+4x+6x^2)^n. Fourth binomial transform of 1/sqrt(1-24x^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).
%C A106258 Also number of paths from (0,0) to (n,0) using steps U=(1,1), H=(1,0) and D=(1,-1), the H steps can have 4 colors and the U steps can have 6 colors. - _N-E. Fahssi_, Mar 31 2008
%H A106258 Vincenzo Librandi, Table of n, a(n) for n = 0..200
%H A106258 Hacène Belbachir, Abdelghani Mehdaoui, László Szalay, Diagonal Sums in the Pascal Pyramid, II: Applications, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.
%H A106258 Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.
%F A106258 E.g.f.: exp(4*x)*BesselI(0, 4*sqrt(3/2)*x); a(n)=sum{k=0..n, C(2k, k)C(k, n-k)2^k}.
%F A106258 D-finite with recurrence: n*a(n) = 4*(2*n-1)*a(n-1) + 8*(n-1)*a(n-2). - _Vaclav Kotesovec_, Oct 17 2012
%F A106258 a(n) ~ sqrt(18+6*sqrt(6))*(4+2*sqrt(6))^n/(6*sqrt(Pi*n)). - _Vaclav Kotesovec_, Oct 17 2012
%t A106258 CoefficientList[Series[1/Sqrt[1-8*x-8*x^2], {x, 0, 20}], x] (* _Vaclav Kotesovec_, Oct 17 2012 *)
%t A106258 RecurrenceTable[{a[0]==1,a[1]==4,a[n]==(4(2n-1)a[n-1]+8(n-1)a[n-2])/n}, a,{n,20}] (* _Harvey P. Dale_, Mar 13 2013 *)
%Y A106258 Cf. A006139, A106259, A106260, A106261.
%K A106258 easy,nonn
%O A106258 0,2
%A A106258 _Paul Barry_, Apr 28 2005
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