A182957 - OEIS

%I #12 Mar 23 2013 02:00:57

%S 1,1,3,17,151,1824,27541,494997,10273039,241217147,6314907390,

%T 182283959604,5750796304553,196865960240416,7268410972604665,

%U 287920792767378837,12181570018235995359,548274960053921957856

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

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

%F G.f. satisfies: [x^n] A(x)^(n+1)/(n+1) = n^n.

%e G.f.: A(x) = 1 + x + 3*x^2 + 17*x^3 + 151*x^4 + 1824*x^5 +...

%e G.f. satisfies A(x) = G(x/A(x)) where A(x*G(x)) = G(x) begins:

%e G(x) = 1 + x + 2^2*x^2 + 3^3*x^3 + 4^4*x^4 + 5^5*x^5 + 6^6*x^6 +...

%e so that:

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

%e The coefficients in A(x)^n for n=1..8 begin:

%e A^1: [(1), 1, 3, 17, 151, 1824, 27541, 494997, ...];

%e A^2: [1,(2), 7, 40, 345, 4052, 59925, 1061154, ...];

%e A^3: [1, 3,(12), 70, 591, 6762, 97938, 1707987, ...];

%e A^4: [1, 4, 18,(108), 899, 10044, 142488, 2446336, ...];

%e A^5: [1, 5, 25, 155,(1280), 14001, 194620, 3288540, ...];

%e A^6: [1, 6, 33, 212, 1746,(18750), 255532, 4248630, ...];

%e A^7: [1, 7, 42, 280, 2310, 24423,(326592), 5342541, ...];

%e A^8: [1, 8, 52, 360, 2986, 31168, 409356, (6588344), ...]; ...

%e where the coefficient of x^n in A(x)^(n+1)/(n+1) equals n^n.

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

%Y  Cf. A180747, A180749.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Dec 31 2010