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A222012 G.f. satisfies: A(x) = Sum_{n>=0} n^n * x^n * A(x)^n / (1 + n*x*A(x))^n. 1

%I

%S 1,1,4,22,145,1081,8863,78751,752587,7708483,84532222,992628616,

%T 12487788067,168344145919,2430351826084,37517872149790,

%U 617842147959019,10821864145358779,200955801421862020,3943205940005194330,81506338541922078355,1769606318933022398611

%N G.f. satisfies: A(x) = Sum_{n>=0} n^n * x^n * A(x)^n / (1 + n*x*A(x))^n.

%F G.f. satisfies: A(x) = 1 + Sum_{n>=1} (n+1)!/2 * x^n * A(x)^n.

%F G.f.: (1/x)*Series_Reversion(x/B(x)), where B(x) = 1 + Sum_{n>=1} (n+1)!/2*x^n.

%e G.f.: A(x) = 1 + x + 4*x^2 + 22*x^3 + 145*x^4 + 1081*x^5 + 8863*x^6 +...

%e where, by definition,

%e A(x) = 1 + x*A(x)/(1+x*A(x)) + 2^2*x^2*A(x)^2/(1+2*x*A(x))^2 + 3^3*x^3*A(x)^3/(1+3*x*A(x))^3 + 4^4*x^4*A(x)^4/(1+4*x*A(x))^4 +....

%e also, g.f. A(x) satisfies:

%e A(x) = 1 + x*A(x) + 3*x^2*A(x)^2 + 12*x^3*A(x)^3 + 60*x^4*A(x)^4 + 360*x^5*A(x)^5 + 2520*x^6*A(x)^6 +...+ (n+1)!/2*x^n*A(x)^n +...

%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, m^m*x^m*A^m/(1+m*x*A+x*O(x^n))^m)); polcoeff(A, n)}

%o for(n=0, 30, print1(a(n), ", "))

%o (PARI) {a(n)=local(B=1+sum(m=1, n, (m+1)!/2*x^m)+x*O(x^n)); polcoeff(1/x*serreverse(x/B), n)}

%o for(n=0, 30, print1(a(n), ", "))

%Y Cf. A211207.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Feb 04 2013

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Last modified July 5 17:29 EDT 2020. Contains 335473 sequences. (Running on oeis4.)