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A098329 Expansion of 1/(1-2x-31x^2)^(1/2). 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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.

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

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 *)

PROG

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

CROSSREFS

Cf. A084603.

Sequence in context: A146706 A120612 A146461 * A160076 A003124 A005570

Adjacent sequences:  A098326 A098327 A098328 * A098330 A098331 A098332

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Sep 03 2004

STATUS

approved

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Last modified June 18 08:52 EDT 2021. Contains 345098 sequences. (Running on oeis4.)