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

%I A098329 #20 Mar 18 2018 11:05:09
%S A098329 1,1,17,49,481,2081,16241,85457,600769,3489601,23391697,143000177,
%T A098329 938797729,5897385313,38397492017,244866166289,1590355308929,
%U A098329 10231490804353,66456634775441,429898281869489,2795449543782241,18150017431150241,118194927388259057,769438418283309649
%N A098329 Expansion of 1/(1-2x-31x^2)^(1/2).
%C A098329 Central coefficient of (1+x+8x^2)^n. 7th binomial transform of 2^n*LegendreP(n,-3) (signed version of A084773).
%C A098329 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
%H A098329 Vincenzo Librandi, <a href="/A098329/b098329.txt">Table of n, a(n) for n = 0..200</a>
%H A098329 Tony D. Noe, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Noe/noe35.html">On the Divisibility of Generalized Central Trinomial Coefficients</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.
%F A098329 a(n) = sum{k=0..floor(n/2), binomial(n-k, k)*binomial(n, k)*8^k}.
%F A098329 E.g.f.: exp(x)*BesselI(0, 4*sqrt(2)*x)
%F A098329 Recurrence: n*a(n) = (2*n-1)*a(n-1) + 31*(n-1)*a(n-2). - _Vaclav Kotesovec_, Oct 14 2012
%F A098329 a(n) ~ sqrt(8+sqrt(2))*(1+4*sqrt(2))^n/(4*sqrt(Pi*n)). - _Vaclav Kotesovec_, Oct 14 2012
%F A098329 a(n) = hypergeom([1/2 - n/2, -n/2], [1], 32). - _Peter Luschny_, Mar 18 2018
%t A098329 Table[SeriesCoefficient[1/Sqrt[1-2*x-31*x^2],{x,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Oct 14 2012 *)
%t A098329 CoefficientList[Series[1/Sqrt[1-2x-31x^2],{x,0,30}],x] (* _Harvey P. Dale_, May 14 2017 *)
%t A098329 a[n_] := Hypergeometric2F1[1/2 - n/2, -n/2, 1, 32];
%t A098329 Table[a[n], {n, 0, 23}] (* _Peter Luschny_, Mar 18 2018 *)
%o A098329 (PARI) x='x+O('x^66); Vec(1/(1-2*x-31*x^2)^(1/2)) \\ _Joerg Arndt_, May 11 2013
%Y A098329 Cf. A084603.
%K A098329 easy,nonn
%O A098329 0,3
%A A098329 _Paul Barry_, Sep 03 2004