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A144684 E.g.f. satisfies: A(x/A(x)^4) = exp(x).

%I A144684 #5 Mar 30 2012 18:37:14
%S A144684 1,1,9,265,15281,1379441,173762425,28528990393,5838405067745,
%T A144684 1443332192692321,421171717332106601,142520112730201819625,
%U A144684 55149333235223148407953,24128837930726025305020369
%N A144684 E.g.f. satisfies: A(x/A(x)^4) = exp(x).
%F A144684 E.g.f. satisfies: A(x) = exp( x*A(log A(x))^4 ).
%F A144684 E.g.f. satisfies: a(n+1) = [x^n/n!] exp(x)*A(x)^(4*n+4).
%F A144684 E.g.f. A(x) = G(4x)^(1/4) where G(x/G(x)) = exp(x) and G(x) is the g.f. of A144681.
%F A144684 E.g.f. A(x) = G(2x)^(1/2) where G(x/G(x)^2) = exp(x) and G(x) is the g.f. of A144682.
%e A144684 E.g.f. A(x) = 1 + x + 9*x^2/2! + 265*x^3/3! + 15281*x^4/4! +...
%e A144684 ILLUSTRATE FORMULA a(n+1) = [x^n/n!] exp(x)*A(x)^(4*n+4) as follows.
%e A144684 Form a table of coefficients of x^k/k! in exp(x)*A(x)^(4*n) for n>=1, k>=0:
%e A144684 exp(x)*A(x)^4: [(1), 5, 57, 1565, 84017, 7220725, 879058921, ...];
%e A144684 exp(x)*A(x)^8: [1, (9), 145, 4377, 231713, 19222569, 2265114033, ...];
%e A144684 exp(x)*A(x)^12:[1, 13,(265), 8821, 472273, 38328733, 4391383897, ...];
%e A144684 exp(x)*A(x)^16:[1, 17, 417,(15281), 841025, 67630417, 7578669793, ...];
%e A144684 exp(x)*A(x)^20:[1, 21, 601, 24141, (1379441), 111109701, 12258211401, ...];
%e A144684 exp(x)*A(x)^24:[1, 25, 817, 35785, 2135137, (173762425), 18997491601, ...];
%e A144684 exp(x)*A(x)^28:[1, 29, 1065, 50597, 3161873, 261721069,(28528990393), ...]; ...
%e A144684 then the terms along the main diagonal form this sequence shift left.
%o A144684 (PARI) {a(n)=local(A=1+x+x*O(x^n));for(n=0,n,A=exp(serreverse(x/A^4)));n!*polcoeff(A,n)}
%o A144684 (PARI) {a(n)=local(A=1+x+sum(k=2, n-1, a(k)*x^k/k!)+x*O(x^n)); if(n==0, 1, (n-1)!*polcoeff(exp(x+x*O(x^n))*A^(4*n), n-1))}
%Y A144684 Cf. A144681, A144682, A144683.
%K A144684 nonn
%O A144684 0,3
%A A144684 _Paul D. Hanna_, Sep 19 2008