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%I #6 Apr 29 2015 10:27:57
%S 1,1,4,60,2480,242296,53763904,28363717952,41396018951936,
%T 215328934357721024,4740698193856769942528,
%U 430771050114778618253200384,151994706469390446336698323709952
%N G.f. satisfies: A(x) = 1 + x*Sum_{n>=0} log( A(2^n*x)^2 )^n/n!.
%F a(n+1) = [x^n] A(x)^(2^(n+1)) for n>=0, with a(0)=1, where A(x) = Sum_{n>=0} a(n)*x^n.
%e G.f.: A(x) = 1 + x + 4*x^2 + 60*x^3 + 2480*x^4 + 242296*x^5 +...
%e A(x) = 1 + x*[1 + log(A(2x)^2) + log(A(4x)^2)^2/2! + log(A(8x)^2)^3/3! + log(A(16x)^2)^4/4! + log(A(32x)^2)^5/5! +...].
%e Coefficients in the 2^n-th powers of A(x) begin:
%e A^(2^0)=[1, 1, 4, 60, 2480, 242296, 53763904, 28363717952,...];
%e A^(2^1)=[(1), 2, 9, 128, 5096, 490032, 108035840, 56837199680,...];
%e A^(2^2)=[1,(4), 22, 292, 10785, 1002752, 218139920, 114116667872,...];
%e A^(2^3)=[1, 8,(60), 760, 24390, 2104632, 444861660, 230028874632,...];
%e A^(2^4)=[1, 16, 184,(2480), 64540, 4690704, 926901832,...];
%e A^(2^5)=[1, 32, 624, 10848,(242296), 12359328, 2033807312,...];
%e A^(2^6)=[1, 64, 2272, 61632, 1568240,(53763904), 5278676128,...];
%e A^(2^7)=[1, 128, 8640, 414080, 16187360, 588318336,(28363717952),...];
%e A^(2^8)=[1, 256, 33664, 3040000, 213028800, 12475903232, 658516757120,(41396018951936),...]; ...
%e where the diagonal terms in parenthesis form this sequence (shift left).
%e The third column of this table is given by A257272 (observation by _Bruno Berselli_). - _M. F. Hasler_, Apr 27 2015
%o (PARI) {a(n)=local(A=1+sum(m=1,n-1,a(m)*x^m));polcoeff(1+x*sum(m=0,n,log(subst(A^2,x,2^m*x)+x*O(x^n))^m/m!),n)}
%Y Cf. A132695, A181445.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Oct 22 2010
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