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%I #4 Jun 14 2018 09:54:39
%S 1,4,8,20,84,456,2860,19708,145120,1122680,9023784,74777248,635292016,
%T 5510485600,48644137764,435920025116,3957758805776,36345636909032,
%U 337159090063880,3155827384249824,29776934546342464,283001546964599248
%N Self-convolution of A004304, where A004304(n) is the number of planar tree-rooted maps with n edges.
%H Vaclav Kotesovec, <a href="/A168451/b168451.txt">Table of n, a(n) for n = 0..840</a>
%F G.f.: A(x) = x/Series_Reversion(x*F(x)^2) where F(x) = g.f. of A005568, where A005568(n) is the product of two successive Catalan numbers C(n)*C(n+1).
%F G.f.: A(x) = F(x/A(x))^2 where A(x*F(x)^2) = F(x)^2 where F(x) = g.f. of A005568.
%e G.f.: A(x) = 1 + 4*x + 8*x^2 + 20*x^3 + 84*x^4 + 456*x^5 + 2860*x^6 +...
%e A(x)^(1/2) = 1 + 2*x + 2*x^2 + 6*x^3 + 28*x^4 + 160*x^5 + 1036*x^6 +...+ A004304(n)*x^n +...
%e G.f. satisfies: A(x*F(x)^2) = F(x)^2 where F(x) = g.f. of A005568:
%e F(x) = 1 + 2*x + 10*x^2 + 70*x^3 + 588*x^4 + 5544*x^5 + 56628*x^6 +...+ A000108(n)*A000108(n+1)*x^n +...
%e F(x)^2 = 1 + 4*x + 24*x^2 + 180*x^3 + 1556*x^4 + 14840*x^5 + 152092*x^6 +...+ A168452(n)*x^n +...
%o (PARI) {a(n)=local(C_2=vector(n+1,m,(binomial(2*m-2,m-1)/m)*(binomial(2*m,m)/(m+1))));polcoeff((x/serreverse(x*Ser(C_2)^2)),n)}
%Y Cf. A168452, A004304, A005568, A000108, variant: A168357.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Nov 26 2009
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