A106258 Expansion of 1/sqrt(1-8x-8x^2).

1, 4, 28, 208, 1624, 13024, 106336, 879232, 7338592, 61699456, 521753728, 4433024512, 37812715264, 323603221504, 2777262164992, 23893731463168, 206005885076992, 1779480850438144, 15396895523989504, 133420304211238912

Offset: 0

Author

Paul Barry, Apr 28 2005

Keywords

• easy
• nonn

Comments

Central coefficient of (1+4x+6x^2)^n. Fourth binomial transform of 1/sqrt(1-24x^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).

Also number of paths from (0,0) to (n,0) using steps U=(1,1), H=(1,0) and D=(1,-1), the H steps can have 4 colors and the U steps can have 6 colors. - N-E. Fahssi, Mar 31 2008

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.
• Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.

Crossrefs

Cf. A006139, A106259, A106260, A106261.

Programs

Mathematica

CoefficientList[Series[1/Sqrt[1-8*x-8*x^2], {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 17 2012 *)
RecurrenceTable[{a[0]==1,a[1]==4,a[n]==(4(2n-1)a[n-1]+8(n-1)a[n-2])/n}, a,{n,20}] (* Harvey P. Dale, Mar 13 2013 *)

Formula

E.g.f.: exp(4*x)*BesselI(0, 4*sqrt(3/2)*x); a(n)=sum{k=0..n, C(2k, k)C(k, n-k)2^k}.

D-finite with recurrence: n*a(n) = 4*(2*n-1)*a(n-1) + 8*(n-1)*a(n-2). - Vaclav Kotesovec, Oct 17 2012

a(n) ~ sqrt(18+6*sqrt(6))*(4+2*sqrt(6))^n/(6*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 17 2012