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%I
%S 1,2,12,86,666,5388,44832,380424,3275172,28512248,250413856,
%T 2215112886,19711078686,176276723508,1583186541144,14271487891512,
%U 129063176166570,1170480053359908,10641805703955624,96970507481607972,885397365149468076,8098908925136867112
%N Self-convolution square-root of A005810, where A005810(n) = binomial(4*n,n).
%F G.f.: A(x) = sqrt( G(x)/(4 - 3*G(x)) ) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293. [From a formula by Mark van Hoeij in A005810]
%e G.f.: A(x) = 1 + 2*x + 12*x^2 + 86*x^3 + 666*x^4 + 5388*x^5 +...
%e The square of the g.f. equals the g.f. of A005810:
%e A(x)^2 = 1 + 4*x + 28*x^2 + 220*x^3 + 1820*x^4 + 15504*x^5 +...
%e The g.f. of A002293 is G(x) = 1 + x*G(x)^4:
%e G(x) = 1 + x + 4*x^2 + 22*x^3 + 140*x^4 + 969*x^5 + 7084*x^6 +...
%o (PARI) {a(n)=polcoeff(sum(k=0,n,binomial(4*k,k)*x^k +x*O(x^n))^(1/2),n)}
%o for(n=0,41,print1(a(n),", "))
%Y Cf. A005810, A002293.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Mar 03 2012
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