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%I #8 Oct 16 2023 01:32:24
%S 1,2,6,56,2176,264128,97403128,116613183904,477185203152432,
%T 6888694977614541952,357723804527747884084384,
%U 67665852938362110551077866496,47032826381397323139718241444226496,120930078672642050250114980899028695276544,1156477264045758740728755778253983148148820652288
%N O.g.f. A(x) satisfies: Sum_{n>=0} (-1)^n * log((1 - 2^n*x)*A(x))^n / n! = 1.
%H Paul D. Hanna, <a href="/A365776/b365776.txt">Table of n, a(n) for n = 0..100</a>
%e G.f.: A(x) = 1 + 2*x + 6*x^2 + 56*x^3 + 2176*x^4 + 264128*x^5 + 97403128*x^6 + 116613183904*x^7 + 477185203152432*x^8 + 6888694977614541952*x^9 + ...
%e where Sum_{n>=0} (-1)^n * log( (1 - 2^n*x)*A(x) )^n / n! = 1.
%e RELATED SERIES.
%e log(A(x)) = 2*x + 8*x^2/2 + 140*x^3/3 + 8264*x^4/4 + 1298472*x^5/5 + 581218736*x^6/6 + 814924372320*x^7/7 + 3815614155894752*x^8/8 + ...
%e log((1-2*x)*A(x)) = 2*x^2 + 44*x^3 + 2062*x^4 + 259688*x^5 + 290609336/3*x^6 + ...
%e log((1-2^2*x)*A(x))^2/2! = 2*x^2 + 8*x^3 - 128/3*x^4 - 12316/3*x^5 + ...
%e log((1-2^3*x)*A(x))^3/3! = -36*x^3 - 504*x^4 - 4584*x^5 - 17204/3*x^6 + ...
%e log((1-2^4*x)*A(x))^4/4! = 4802/3*x^4 + 170128/3*x^5 + 12208448/9*x^6 + ...
%e log((1-2^5*x)*A(x))^5/5! = -202500*x^5 - 17145000*x^6 - 947709000*x^7 + ...
%e ...
%o (PARI) {a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); A[#A] = Vec(sum(n=0, #A+1, (-1)^n * log( (1 - 2^n*x)*Ser(A) )^n/n! ))[#A] ); A[n+1]}
%o for(n=0, 20, print1(a(n), ", "))
%Y Cf. A306062.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Oct 15 2023
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