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A111993 Fifth convolution of Schroeder's (second problem) numbers A001003(n), n>=0.

%I A111993 #13 Mar 17 2017 00:47:12
%S A111993 1,5,25,125,630,3206,16470,85350,445775,2344595,12408903,66042795,
%T A111993 353259900,1898119100,10240583420,55454182716,301307002605,
%U A111993 1642192132625,8975693643525,49186242980105,270186765784210
%N A111993 Fifth convolution of Schroeder's (second problem) numbers A001003(n), n>=0.
%H A111993 Vincenzo Librandi, <a href="/A111993/b111993.txt">Table of n, a(n) for n = 0..300</a>
%F A111993 G.f.: ((1+x-sqrt(1-6*x+x^2))/(4*x))^5.
%F A111993 a(n)= (5/n)*Sum_{k=1,..,n} binomial(n,k)*binomial(n+k+4,k-1), a(0)=1.
%F A111993 a(n) = 5*hypergeom([1-n, n+6], [2], -1), n>=1, a(0)=1.
%F A111993 Recurrence: n*(n+5)*a(n) = n*(7*n+23)*a(n-1) - (n+2)*(7*n-9)*a(n-2) + (n-3)*(n+2)*a(n-3). - _Vaclav Kotesovec_, Oct 18 2012
%F A111993 a(n) ~ 5*sqrt(3*sqrt(2)-4)*(17-12*sqrt(2)) * (3+2*sqrt(2))^(n+5)/(16*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 18 2012
%t A111993 CoefficientList[Series[((1+x-Sqrt[1-6*x+x^2])/(4*x))^5, {x, 0, 20}], x] (* _Vaclav Kotesovec_, Oct 18 2012 *)
%o A111993 (PARI) x='x+O('x^50); Vec(((1+x-sqrt(1-6*x+x^2))/(4*x))^5) \\ _G. C. Greubel_, Mar 16 2017
%Y A111993 Cf. Fifth column of convolution triangle A011117. Fourth convolution: A010849.
%K A111993 nonn,easy
%O A111993 0,2
%A A111993 _Wolfdieter Lang_, Sep 12 2005