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%I #14 Jul 04 2014 02:41:16
%S 1,20,3850,1078000,355066250,128107903000,49001272897500,
%T 19520507080800000,8012558140822125000,3365274419145292500000,
%U 1439327869068441602250000,624739666805574817770000000
%N a(n) = (5^n/n!^2) * Product_{k=0..n-1} (10k+1)*(10k+4).
%H Vincenzo Librandi, <a href="/A184891/b184891.txt">Table of n, a(n) for n = 0..100</a>
%F Self-convolution yields Sum_{k=0..n} a(n-k)*a(k) = A184892(n) where
%F . A184892(n) = C(2n,n) * (5^n/n!^2)*Product_{k=0..n-1} (5k+1)*(5k+4).
%e G.f.: A(x) = 1 + 20*x + 3850*x^2 + 1078000*x^3 +...
%e A(x)^2 = 1 + 40*x + 8100*x^2 + 2310000*x^3 +...+ A184892(n)*x^n +...
%t Table[5^n/(n!)^2 Product[(10k+1)(10k+4),{k,0,n-1}],{n,0,20}] (* _Harvey P. Dale_, Feb 02 2012 *)
%t FullSimplify[Table[2^(2*n) * 5^(3*n) * Gamma[n+1/10] * Gamma[n+2/5] / (Gamma[2/5] * Gamma[1/10] * Gamma[n+1]^2), {n, 0, 15}]] (* _Vaclav Kotesovec_, Jul 03 2014 *)
%o (PARI) {a(n)=(5^n/n!^2)*prod(k=0,n-1,(10*k+1)*(10*k+4))}
%Y Cf. A184892; variants: A184424, A178529, A092870, A184895, A184897.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jan 25 2011
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