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%I #19 Mar 17 2017 00:47:30
%S 1,8,52,312,1802,10200,57092,317544,1760035,9738160,53844184,
%T 297717712,1646893140,9116815952,50514367512,280173703472,
%U 1555632093093,8647009926904,48117998453036,268057662257096,1494927614877214
%N Eighth convolution of Schroeder's (second problem) numbers A001003(n), n>=0.
%H Vincenzo Librandi, <a href="/A111996/b111996.txt">Table of n, a(n) for n = 0..300</a>
%F G.f.: ((1+x-sqrt(1-6*x+x^2))/(4*x))^8.
%F a(n)= (8/n)*Sum_{k=1,..,n} binomial(n,k)*binomial(n+k+7,k-1).
%F a(n) = 8*hypergeom([1-n, n+9], [2], -1), n>=1, a(0)=1.
%F Recurrence: n*(n+8)*a(n) = (7*n^2+44*n+21)*a(n-1) - (7*n^2+26*n-24)*a(n-2) + (n-3)*(n+5)*a(n-3). - _Vaclav Kotesovec_, Oct 18 2012
%F a(n) ~ sqrt(3*sqrt(2)-4)*(338-239*sqrt(2)) * (3+2*sqrt(2))^(n+8)/(8*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 18 2012
%t CoefficientList[Series[((1+x-Sqrt[1-6x+x^2])/(4x))^8, {x,0,20}], x] (* _Harvey P. Dale_, Apr 01 2011 *)
%o (PARI) x='x+O(x^50); Vec(((1+x-sqrt(1-6*x+x^2))/(4*x))^8) \\ _G. C. Greubel_, Mar 16 2017
%Y Cf. Eighth column of convolution triangle A011117.
%K nonn,easy
%O 0,2
%A _Wolfdieter Lang_, Sep 12 2005
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