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%I #7 Sep 11 2013 19:21:31
%S 1,1,6,44,430,5162,72890,1178978,21449704,433116488,9606793874,
%T 232145293502,6070097785376,170763070370848,5142963967765530,
%U 165115679014587758,5629558857460143814,203146937778126705662,7735490130309647256862
%N G.f.: ( Sum_{n>=0} (2*n+x)^n * x^n / (1 + 2*n*x + x^2)^n )^(1/2).
%F Self-convolution yields A187746.
%F Sum_{k=0..n} a(n)*a(n-k) = (2*n^2 + 2*n + 1) * 2^(n-2) * (n-1)! for n>1.
%e G.f.: A(x) = 1 + x + 6*x^2 + 44*x^3 + 430*x^4 + 5162*x^5 +...
%e where the square of the g.f. begins
%e A(x)^2 = 1 + 2*x + 13*x^2 + 100*x^3 + 984*x^4 + 11712*x^5 + 163200*x^6 +...+ A187746(n)*x^n +...
%e and equals the series
%e A(x)^2 = 1 + (2+x)*x/(1+2*x+x^2) + (4+x)^2*x^2/(1+4*x+x^2)^2 + (6+x)^3*x^3/(1+6*x+x^2)^3 + (8+x)^4*x^4/(1+8*x+x^2)^4 + (10+x)^5*x^5/(1+10*x+x^2)^5 +...
%o (PARI) {a(n)=polcoeff( sum(m=0, n, (2*m+x)^m*x^m/(1+2*m*x+x^2 +x*O(x^n))^m)^(1/2), n)}
%o for(n=0, 30, print1(a(n), ", "))
%K nonn
%O 0,3
%A _Paul D. Hanna_, May 24 2013
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