A098455 Expansion of 1/sqrt(1-4*x-36*x^2).

1, 2, 24, 128, 1096, 7632, 60864, 461568, 3648096, 28551872, 226695424, 1799989248, 14380907776, 115126211072, 924791445504, 7444100947968, 60057602459136, 485388465196032, 3929580292706304, 31858982479331328, 258641677679947776, 2102242140708298752

Offset: 0

Author

Paul Barry, Sep 08 2004

Keywords

• easy
• nonn
• changed

Comments

Define Q(n,x) = Sum_{k=0..floor(n/2)} binomial(n,k) * binomial(2(n-k),n) * x^(n-2k). Then a(n) = 3^n*Q(n,1/3). A084770(n) is 2^n*Q(n,1/2). Central coefficient of (1+2*x+10*x^2)^n.

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.

Crossrefs

Cf. A387428.

Programs

Mathematica

Table[SeriesCoefficient[1/Sqrt[1-4*x-36*x^2],{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 15 2012 *)

PARI

x='x+O('x^66); Vec(1/sqrt(1-4*x-36*x^2)) \\ Joerg Arndt, May 11 2013

Formula

E.g.f.: exp(2*x) * BesselI(0, 2*sqrt(10)*x).

a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)*binomial(2(n-k), n)*9^k.

D-finite with recurrence: n*a(n) = 2*(2*n-1)*a(n-1) + 36*(n-1)*a(n-2). - Vaclav Kotesovec, Oct 15 2012

a(n) ~ sqrt(50+5*sqrt(10))*(2+2*sqrt(10))^n/(10*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 15 2012

From Seiichi Manyama, Aug 30 2025: (Start)

a(n) = Sum_{k=0..n} (1-3*i)^k * (1+3*i)^(n-k) * binomial(n,k)^2, where i is the imaginary unit.

a(n) = Sum_{k=0..floor(n/2)} 10^k * 2^(n-2*k) * binomial(n,2*k) * binomial(2*k,k). (End)