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%I #6 Sep 13 2014 18:56:11
%S 1,2,15,234,6019,226656,11629128,774788698,64757369211,6615393335250,
%T 809323719822671,116638942433360112,19535480098041792024,
%U 3759317862736434388304,823134193681237065635088,203355215614514847510001434,56269314099500094422938613707
%N G.f.: 1 = Sum_{n>=0} a(n)*x^n * [ Sum_{k=0..n+1} binomial(n+1, k)^2*(-x)^k ]^2.
%C Compare to the g.f. G(x) for A006013(n) = C(3*n+1,n)/(n+1), which satisfies:
%C (1) 1 = Sum_{n>=0} A006013(n)*x^n*[Sum_{k=0..n+1} C(n+1,k)^2*(-x)^k]^2,
%C (2) G(x) = (1 + x*G(x)^(3/2))^2 so that G(x)^(1/2) is an integer series.
%e G.f.: A(x) = 1 + 2*x + 15*x^2 + 234*x^3 + 6019*x^4 + 226656*x^5 +...
%e Note that the square-root of the g.f., A(x)^(1/2), is an integer series:
%e A(x)^(1/2) = 1 + x + 7*x^2 + 110*x^3 + 2875*x^4 + 109683*x^5 +...+ A212371(n)*x^n +...
%o (PARI) {a(n)=if(n==0, 1, -polcoeff(sum(m=0, n-1, a(m)*x^m*sum(k=0, m+1, binomial(m+1, k)^2*(-x)^k)^2+x*O(x^n)), n))}
%o for(n=0,20,print1(a(n),", "))
%Y Cf. A212371, A247031.
%K nonn
%O 0,2
%A _Paul D. Hanna_, May 10 2012
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