A106260 Expansion of 1/sqrt(1-16x-16x^2).

1, 8, 104, 1472, 21856, 333568, 5183744, 81590272, 1296426496, 20750839808, 334081306624, 5404163080192, 87763693060096, 1430025994108928, 23367175920287744, 382767375745810432, 6283401962864377856

Offset: 0

Author

Paul Barry, Apr 28 2005

Keywords

• easy
• nonn

Comments

Central coefficient of (1+8x+20x^2)^n. Eighth binomial transform of 1/sqrt(1-80x^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.

Crossrefs

Cf. A006139, A106258, A106259, A106261.

Programs

Mathematica

CoefficientList[Series[1/Sqrt[1-16*x-16*x^2], {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 17 2012 *)

Formula

E.g.f.: exp(8*x)*BesselI(0, 8*sqrt(5/4)*x); a(n)=sum{k=0..n, C(2k, k)C(k, n-k)4^k}.

D-finite with recurrence: n*a(n) = 8*(2*n-1)*a(n-1) + 16*(n-1)*a(n-2). - Vaclav Kotesovec, Oct 17 2012

a(n) ~ sqrt(50+20*sqrt(5))*(8+4*sqrt(5))^n/(10*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 17 2012