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%I #14 Mar 17 2017 00:47:38
%S 1,9,63,399,2403,14067,80949,460845,2605590,14666470,82320714,
%T 461238282,2581644378,14442658074,80785970838,451934259654,
%U 2528977211775,14157983986839,79302044283297,444448115168049,2492468172937125
%N Ninth convolution of Schroeder's (second problem) numbers A001003(n), n>=0.
%H Vincenzo Librandi, <a href="/A111997/b111997.txt">Table of n, a(n) for n = 0..200</a>
%F G.f.: ((1+x-sqrt(1-6*x+x^2))/(4*x))^9.
%F a(n) = (9/n)*Sum_{k=1,..,n} binomial(n,k)*binomial(n+k+8,k-1).
%F a(n) = 9*hypergeom([1-n, n+10], [2], -1), n>=1, a(0)=1.
%F Recurrence: n*(n+9)*a(n) = (7*n^2+51*n+32)*a(n-1) - (7*n^2+33*n-22)*a(n-2) + (n-3)*(n+6)*a(n-3). - _Vaclav Kotesovec_, Oct 18 2012
%F a(n) ~ 9*sqrt(3*sqrt(2)-4)*(577-408*sqrt(2)) * (3+2*sqrt(2))^(n+9)/(64*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 18 2012
%t CoefficientList[Series[((1+x-Sqrt[1-6*x+x^2])/(4*x))^9, {x, 0, 20}], x] (* _Vaclav Kotesovec_, Oct 18 2012 *)
%o (PARI) x='x+O('x^50); Vec(((1+x-sqrt(1-6*x+x^2))/(4*x))^9) \\ _G. C. Greubel_, Mar 17 2017
%Y Ninth column of convolution triangle A011117.
%K nonn,easy
%O 0,2
%A _Wolfdieter Lang_, Sep 12 2005
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