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%I #11 Jun 01 2014 20:25:11
%S 1,1,3,23,373,10625,468449,29464867,2501594061,275449029949,
%T 38159974444647,6494568101890209,1331919152059028501,
%U 323931747385690406859,92181008940950546725329,30343276257278261997917167,11439766584058218615650332597,4897281474298488766530388729025
%N G.f.: A(x) = F(0,x) where F(0,x) = 1/(1 - x*F(1,x)), F(1,x) = 1/(1 - 4*x*F(2,x)^2)^(1/2), F(2,x) = 1/(1 - 9*x/F(3,x)^3)^(1/3), F(3,x) = 1/(1 - 16*x/F(4,x)^4)^(1/4), ..., so that F(n,x) = 1/(1 - (n+1)^2*x*F(n+1,x)^(n+1))^(1/(n+1)) for n>=0.
%C Compare to an o.g.f. E(x) for the Euler numbers (A000364):
%C E(x) = G(0,x) where G(0,x) = 1/(1 - x*G(1,x)^2), G(1,x) = 1/(1 - 4*x*G(2,x)^3)^(1/2), G(2,x) = 1/(1 - 9*x/G(3,x)^4)^(1/3), G(3,x) = 1/(1 - 16*x/G(4,x)^5)^(1/4), ..., so that G(n,x) = 1/(1 - (n+1)^2*x*G(n+1,x)^(n+2))^(1/(n+1)) for n>=0.
%H Paul D. Hanna, <a href="/A243195/b243195.txt">Table of n, a(n) for n = 0..150</a>
%e G.f.: A(x) = 1 + x + 3*x^2 + 17*x^3 + 151*x^4 + 1901*x^5 + 31841*x^6 +...
%e Define F(n,x) with F(0,x) = A(x) where for n>0 we have
%e F(n-1,x) = 1/(1 - n^2*x*F(n,x)^n)^(1/n), or equivalently
%e F(n,x) = ( (1 - 1/F(n-1,x)^n) / (n^2*x) )^(1/n),
%e then
%e F(1,x) = 1 + 2*x + 18*x^2 + 326*x^3 + 9826*x^4 + 445860*x^5 +...
%e F(2,x) = 1 + 3*x + 54*x^2 + 1782*x^3 + 88113*x^4 + 6006267*x^5 +...
%e F(3,x) = 1 + 4*x + 120*x^2 + 6280*x^3 + 465040*x^4 + 45026364*x^5 +...
%e F(4,x) = 1 + 5*x + 225*x^2 + 17125*x^3 + 1779800*x^4 + 233701800*x^5 +...
%e F(5,x) = 1 + 6*x + 378*x^2 + 39438*x^3 + 5484738*x^4 + 941166954*x^5 +...
%e F(6,x) = 1 + 7*x + 588*x^2 + 80556*x^3 + 14457646*x^4 + 3147092914*x^5 +...
%e F(7,x) = 1 + 8*x + 864*x^2 + 150432*x^3 + 33866688*x^4 + 9128319696*x^5 +...
%e ...
%e F(1,x)^2 = 1 + 4*x + 40*x^2 + 724*x^3 + 21280*x^4 + 942760*x^5 +...
%e F(2,x)^3 = 1 + 9*x + 189*x^2 + 6345*x^3 + 306621*x^4 + 20256561*x^5 +...
%e F(3,x)^4 = 1 + 16*x + 576*x^2 + 31136*x^3 + 2271296*x^4 + 213398256*x^5 +...
%e F(4,x)^5 = 1 + 25*x + 1375*x^2 + 109375*x^3 + 11289625*x^4 + 1444554625*x^5 +...
%e F(5,x)^6 = 1 + 36*x + 2808*x^2 + 308988*x^3 + 42986448*x^4 + 7223051340*x^5 +...
%e F(6,x)^7 = 1 + 49*x + 5145*x^2 + 748769*x^3 + 135256905*x^4 + 28966780465*x^5 +...
%e ...
%o (PARI) {a(n)=local(A=1+n*x); for(k=0, n-1, A=(1 - (n-k)^2*x*A^(n-k) +x*O(x^n))^(-1/(n-k))); polcoeff(A, n)}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A075820, A243161.
%K nonn
%O 0,3
%A _Paul D. Hanna_, May 31 2014
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