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Expansion of 1/sqrt(1-12x-12x^2).
5

%I #17 Jan 30 2020 21:29:15

%S 1,6,60,648,7344,85536,1014336,12182400,147702528,1803907584,

%T 22159733760,273508669440,3389106769920,42134712606720,

%U 525323149885440,6565657319866368,82235651779657728,1031956779869798400

%N Expansion of 1/sqrt(1-12x-12x^2).

%C Central coefficient of (1+6x+12x^2)^n. Sixth binomial transform of 1/sqrt(1-48x^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="/A106259/b106259.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(6*x)*BesselI(0, 6*sqrt(4/3)*x); a(n)=sum{k=0..n, C(2k, k)C(k, n-k)3^k}.

%F D-finite with recurrence: n*a(n) = 6*(2*n-1)*a(n-1) + 12*(n-1)*a(n-2). - _Vaclav Kotesovec_, Oct 17 2012

%F a(n) ~ (1+sqrt(3))*(6+4*sqrt(3))^n/(2*sqrt(2*Pi*n)). - _Vaclav Kotesovec_, Oct 17 2012

%t CoefficientList[Series[1/Sqrt[1-12*x-12*x^2], {x, 0, 20}], x] (* _Vaclav Kotesovec_, Oct 17 2012 *)

%Y Cf. A006139, A106258, A106260, A106261.

%K easy,nonn

%O 0,2

%A _Paul Barry_, Apr 28 2005