A258723 Expansion of 1/(1-12*x+48*x^2)^(1/2).

1, 6, 30, 108, 54, -3564, -41364, -314280, -1798362, -6972156, -1793340, 283697640, 3341429820, 25984971720, 151750943640, 596184213168, 101849014278, -25747257110940, -305001821608236, -2392882855430328, -14088646343199276, -55649498057805096, -7100681134947480

Offset: 0

Author

Sergei N. Gladkovskii, Jun 08 2015

Keywords

• sign

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Hacène Belbachir, Abdelghani Mehdaoui, László Szalay, Diagonal Sums in the Pascal Pyramid, II: Applications, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.
• Vaclav Kotesovec, Graph - the asymptotic ratio

Crossrefs

Cf. A144535, A144536, A240880.

Programs

Mathematica

CoefficientList[Series[1/(1-12*x+48*x^2)^(1/2), {x, 0, 20}], x]  (* Vaclav Kotesovec, Jun 08 2015 *)

PARI

Vec(1/(1-12*x+48*x^2)^(1/2) + x^50) \\ G. C. Greubel, Feb 14 2017

Formula

G.f.: 1/(1-12*x+48*x^2)^(1/2).

E.g.f.: exp(6*x)*BesselJ(0,2*sqrt(3)*x).

If mod(n,6)=4 then a(n) ~ (-1)^((n+8)/6) * 3^((n+1)/2) * 4^(n-1) / (sqrt(Pi) * n^(3/2)), else a(n) ~ 3^(n/2) * 2^(2*n+1) * cos(Pi*(n-1)/6) / sqrt(Pi*n). - Vaclav Kotesovec, Jun 08 2015

D-finite with recurrence n*a(n) +6*(-2*n+1)*a(n-1) +48*(n-1)*a(n-2)=0. [Belbachir]