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%I #11 Jun 16 2015 19:29:32
%S 1,2,12,192,4272,124992,4531392,195869952,9832326912,562125837312,
%T 36056880110592,2564230500421632,200237330428342272,
%U 17032391106795159552,1567547894591436275712,155196096043697480466432,16447362605632117421309952,1857733260790463501532659712
%N E.g.f.: A'(x) = 1 + A(x)^6, with A(0)=1.
%C In general, for k>1, if e.g.f. satisfies A'(x) = 1 + A(x)^k, with A(0)=1, then a(n) ~ n! * d^(n + 1/(k-1)) / ((k-1)^(1/(k-1)) * Gamma(1/(k-1)) * n^(1-1/(k-1))), where d = 1 / Sum_{j>=1} (-1)^(j+1)/(k*j-1).
%H Vaclav Kotesovec, <a href="/A258994/b258994.txt">Table of n, a(n) for n = 0..50</a>
%F a(n) ~ n! * d^(n+1/5) / (5^(1/5) * Gamma(1/5) * n^(4/5)), where d = 1 / Sum_{j>=1} (-1)^(j+1)/(6*j-1) = 6/(Pi - sqrt(3)*log(2+sqrt(3))) = 6.97224737278326506475991855023425659249063565...
%F E.g.f.: 1 + Series_Reversion( Integral 1/(1 + (1+x)^6) dx ). - _Paul D. Hanna_, Jun 16 2015
%e A(x) = 1 + 2*x + 12*x^2/2! + 192*x^3/3! + 4272*x^4/4! + 124992*x^5/5! + ...
%e A'(x) = 2 + 12*x + 96*x^2 + 712*x^3 + 5208*x^4 + 188808*x^5/5 + ...
%e 1 + A(x)^6 = 2 + 12*x + 96*x^2 + 712*x^3 + 5208*x^4 + 188808*x^5/5 + ...
%t nmax=20; Subscript[a,0]=1; egf=Sum[Subscript[a,k]*x^k,{k,0,nmax+1}]; Table[Subscript[a,k]*k!,{k,0,nmax}] /.Solve[Take[CoefficientList[Expand[1+egf^6-D[egf,x]],x],nmax]==ConstantArray[0,nmax]][[1]]
%o (PARI) {a(n) = local(A=1); A = 1 + serreverse( intformal( 1/(1 + (1+x)^6 +x*O(x^n)) )); n!*polcoeff(A, n)}
%o for(n=0, 25, print1(a(n), ", ")) \\ _Paul D. Hanna_, Jun 16 2015
%Y Cf. A000831 (k=2), A258969 (k=3), A258970 (k=4), A258971 (k=5), A258927.
%K nonn
%O 0,2
%A _Vaclav Kotesovec_, Jun 16 2015
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