A106259 Expansion of 1/sqrt(1-12x-12x^2).

1, 6, 60, 648, 7344, 85536, 1014336, 12182400, 147702528, 1803907584, 22159733760, 273508669440, 3389106769920, 42134712606720, 525323149885440, 6565657319866368, 82235651779657728, 1031956779869798400

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

Cf. A006139, A106258, A106260, A106261.

Sequence in context: A186674 A186672 A295503 * A085364 A228484 A232969

Adjacent sequences: A106256 A106257 A106258 * A106260 A106261 A106262

Keyword

easy,nonn

Author

Paul Barry, Apr 28 2005

Status

approved