A098329 Expansion of 1/(1-2x-31x^2)^(1/2).

1, 1, 17, 49, 481, 2081, 16241, 85457, 600769, 3489601, 23391697, 143000177, 938797729, 5897385313, 38397492017, 244866166289, 1590355308929, 10231490804353, 66456634775441, 429898281869489, 2795449543782241, 18150017431150241, 118194927388259057, 769438418283309649

Offset: 0

Author

Paul Barry, Sep 03 2004

Keywords

• easy
• nonn

Comments

Central coefficient of (1+x+8x^2)^n. 7th binomial transform of 2^n*LegendreP(n,-3) (signed version of A084773).

Also number of paths from (0,0) to (n,0) using steps U=(1,1), H=(1,0) and D=(1,-1), the U steps can have 8 colors. - N-E. Fahssi, Mar 31 2008

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.

Crossrefs

Cf. A084603.

Programs

Mathematica

Table[SeriesCoefficient[1/Sqrt[1-2*x-31*x^2],{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 14 2012 *)
CoefficientList[Series[1/Sqrt[1-2x-31x^2],{x,0,30}],x] (* Harvey P. Dale, May 14 2017 *)
a[n_] := Hypergeometric2F1[1/2 - n/2, -n/2, 1, 32];
Table[a[n], {n, 0, 23}] (* Peter Luschny, Mar 18 2018 *)

PARI

x='x+O('x^66); Vec(1/(1-2*x-31*x^2)^(1/2)) \\ Joerg Arndt, May 11 2013

Formula

a(n) = sum{k=0..floor(n/2), binomial(n-k, k)*binomial(n, k)*8^k}.

E.g.f.: exp(x)*BesselI(0, 4*sqrt(2)*x)

Recurrence: n*a(n) = (2*n-1)*a(n-1) + 31*(n-1)*a(n-2). - Vaclav Kotesovec, Oct 14 2012

a(n) ~ sqrt(8+sqrt(2))*(1+4*sqrt(2))^n/(4*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 14 2012

a(n) = hypergeom([1/2 - n/2, -n/2], [1], 32). - Peter Luschny, Mar 18 2018