editing

approved

E.g.f.: A(x) = exp( x/A(-x*A(x)^6)^2 ).

Compare the e.g.f. to:

E.g.f.: A(x) = 1 + x + 5*x^2/2! + 73*x^3/3! + 1497*x^4/4! + 48321*x^5/5! +...

The logarithm of the e.g.f., log(A(x) ) = x/A(-x*A(x)^6)^2, begins:

log(A(x) ) = x + 4*x^2/2! + 60*x^3/3! + 1184*x^4/4! + 39120*x^5/5! + 1639872*x^6/6! +...

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=exp(x/subst(A^2, x, -x*A^6+x*O(x^n)))); n!*polcoeff(A, n)}

Cf. A213108, A213109, A213110, A213112, A213113.

approved

editing

editing

approved

allocated for Paul D. Hanna

E.g.f.: A(x) = exp( x/A(-x*A(x)^6)^2 ).

1, 1, 5, 73, 1497, 48321, 2016733, 106687113, 6745180529, 495988880833, 41495596689141, 3880618840698249, 400537444634948041, 45126092520882513921, 5501154522933362385485, 720279890636684703825481, 100658531630809161730405857, 14934726665907895887483076737

0,3

Compare the e.g.f. to:

(1) W(x) = exp(x/W(-x*W(x)^2)^1) when W(x) = Sum_{n>=0} (1*n+1)^(n-1)*x^n/n!.

(2) W(x) = exp(x/W(-x*W(x)^4)^2) when W(x) = Sum_{n>=0} (2*n+1)^(n-1)*x^n/n!.

(3) W(x) = exp(x/W(-x*W(x)^6)^3) when W(x) = Sum_{n>=0} (3*n+1)^(n-1)*x^n/n!.

E.g.f.: A(x) = 1 + x + 5*x^2/2! + 73*x^3/3! + 1497*x^4/4! + 48321*x^5/5! +...

Related expansions:

A(x)^2 = 1 + 2*x + 12*x^2/2! + 176*x^3/3! + 3728*x^4/4! + 118912*x^5/5! +...

A(x)^6 = 1 + 6*x + 60*x^2/2! + 1008*x^3/3! + 23952*x^4/4! + 775296*x^5/5! +...

1/A(-x*A(x)^6)^2 = 1 + 2*x + 20*x^2/2! + 296*x^3/3! + 7824*x^4/4! +...

The logarithm of the e.g.f., log(A(x) = x/A(-x*A(x)^6)^2, begins:

log(A(x) = x + 4*x^2/2! + 60*x^3/3! + 1184*x^4/4! + 39120*x^5/5! + 1639872*x^6/6! +...

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=exp(x/subst(A^2, x, -x*A^6+x*O(x^n)))); n!*polcoeff(A, n)}

for(n=0, 25, print1(a(n), ", "))

Cf. A213108, A213109, A213110, A213112, A213113.

allocated

nonn

Paul D. Hanna, Jun 05 2012

approved

editing

allocated for Paul D. Hanna

allocated

approved