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%I #12 Mar 19 2016 09:58:24
%S 1,1,4,28,276,3480,53232,955524,19672320,456803328,11810032896,
%T 336463895808,10473959755008,353739038360832,12883270796310528,
%U 503352328766459904,21001144899441162240,931963581151516477440,43832663421577452887040,2178029362561822117094400,114014865901176834809333760
%N G.f.: A(x) = 1+x*(1+2x*(1+3x*(...(1+n*x*(...)^2)^2...)^2)^2)^2.
%H Paul D. Hanna and Vaclav Kotesovec, <a href="/A128318/b128318.txt">Table of n, a(n) for n = 0..385</a> (terms 0..200 from Paul D. Hanna)
%F Conjecture: a(n) ~ n! * (8/3)^n / sqrt(n). - _Vaclav Kotesovec_, Mar 19 2016
%e G.f.: A(x) = 1 + x*B(x)^2; B(x) = 1 + 2*x*C(x)^2; C(x) = 1 + 3*x*D(x)^2; D(x) = 1 + 4*x*E(x)^2; E(x) = 1 + 5*x*F(x)^2; F(x) = 1 + 6*x*G(x)^2; ...
%e where the respective sequences begin:
%e A=[1,1,4,28,276,3480,53232,955524,19672320,...];
%e B=[1,2,12,114,1440,22368,409248,8585088,202733760,...];
%e C=[1,3,24,288,4440,82080,1752000,42178800,1127335680,...];
%e D=[1,4,40,580,10560,226560,5532960,150570240,4501422240,...];
%e E=[1,5,60,1020,21420,523320,14399280,437433780,14479664640,...];
%e F=[1,6,84,1638,38976,1068480,32716992,1098069504,39896236800,...];
%e G=[1,7,112,2464,65520,1991808,67189248,2469837888,97765355520,...];
%e H=[1,8,144,3528,103680,3461760,127569600,5098406400,218459165760,...];
%o (PARI) {a(n)=local(A=1+(n+1)*x);for(k=0,n,A=1+(n-k+1)*x*A^2 +x*O(x^n));polcoeff(A,n)}
%o for(n=0, 25, print1(a(n), ", "))
%Y Cf. A128319, A268652.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Mar 07 2007
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