OFFSET
0,2
COMMENTS
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).
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
Hacène Belbachir, Abdelghani Mehdaoui, László Szalay, Diagonal Sums in the Pascal Pyramid, II: Applications, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.
FORMULA
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}.
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
a(n) ~ (1+sqrt(3))*(6+4*sqrt(3))^n/(2*sqrt(2*Pi*n)). - Vaclav Kotesovec, Oct 17 2012
MATHEMATICA
CoefficientList[Series[1/Sqrt[1-12*x-12*x^2], {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 17 2012 *)
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Apr 28 2005
STATUS
approved