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A115902 Expansion of (1-8*x)^(-3/2).

%I A115902 #33 Jan 27 2024 07:05:18
%S A115902 1,12,120,1120,10080,88704,768768,6589440,56010240,472975360,
%T A115902 3972993024,33228668928,276905574400,2300446310400,19060840857600,
%U A115902 157569617756160,1299949346488320,10705465206374400,88022713919078400,722712809019801600,5926245033962373120
%N A115902 Expansion of (1-8*x)^(-3/2).
%H A115902 Youngja Park and SeungKyung Park, <a href="http://dx.doi.org/10.1016/j.disc.2016.04.024">Enumeration of generalized lattice paths by string types, peaks, and ascents</a>, Discrete Mathematics 339.11 (2016): 2652-2659.
%F A115902 G.f.: 1/((1-8*x)*sqrt(1-8*x)) = 1F0(3/2;;8x).
%F A115902 a(n) = Jacobi_P(n,1/2,1/2,1)*8^n.
%F A115902 a(n) = 2^n*(2*n+1)*binomial(2*n,n).
%F A115902 a(n) = (2*n+1)*A059304(n).
%F A115902 a(n) = 2^n*A002457(n).
%F A115902 D-finite with recurrence: n*a(n) -4*(2*n+1)*a(n-1) =0. - _R. J. Mathar_, Nov 14 2011
%F A115902 G.f.: G(0)/2, where G(k) = 1 + 1/(1 - 8*x*(2*k+3)/(8*x*(2*k+3) + 2*(k+1)/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Jun 06 2013
%F A115902 Sum_{n>=0} (-1)^n/a(n) = 4/3*log(2). - _Daniel Suteu_, Oct 31 2017
%F A115902 Sum_{n>=0} 1/a(n) = 8*arcsin(1/sqrt(8))/sqrt(7). - _Amiram Eldar_, Jan 27 2024
%p A115902 a:= n-> add((binomial(2*n,n))*2^(n-2), j=1..n): seq(a(n), n=1..20); # _Zerinvary Lajos_, May 03 2007
%t A115902 CoefficientList[Series[(1-8x)^-(3/2),{x,0,30}],x] (* _Harvey P. Dale_, Jul 13 2012 *)
%Y A115902 Cf. A002457, A059304.
%K A115902 easy,nonn
%O A115902 0,2
%A A115902 _Paul Barry_, Feb 02 2006