A188194 -id:A188194

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A188193

G.f. satisfies: A(x) = Sum_{n>=0} log(1 + 2^n*x*A(x))^n/n!.

+0
1

1, 2, 10, 100, 2500, 224728, 77611032, 95603336016, 411188458873152, 6215509773143124736, 334390128406134844422816, 64839530694681966290325813952, 45813418110052719651124682371286592

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OFFSET

0,2

LINKS

Table of n, a(n) for n=0..12.

FORMULA

G.f. A(x) satisfies:

(1) A(x) = Sum_{n>=0} C(2^n,n)*x^n*A(x)^n,

(2) A(x) = (1/x)*Series_Reversion(x/B(x)),

(3) A(x) = B(x*A(x)) and B(x) = A(x/B(x)),

where B(x) = Sum_{n>=0} C(2^n,n)*x^n is the g.f. of A014070.

(4) A(x) = G(x/A(x)) and G(x) = A(x*G(x)), where G(x) is the g.f. of A188194.

EXAMPLE

G.f.: A(x) = 1 + 2*x + 10*x^2 + 100*x^3 + 2500*x^4 + 224728*x^5 +...

which equals the series:

A(x) = 1 + log(1+2*x*A(x)) + log(1+4*x*A(x))^2/2! + log(1+8*x*A(x))^3/3! +...

Let B(x) equal the g.f. of A014070, which begins:

B(x) = 1 + 2*x + 6*x^2 + 56*x^3 + 1820*x^4 +...+ C(2^n,n)*x^n +...

then B(x) = A(x/B(x)) and A(x) = B(x*A(x)), so that:

A(x) = 1 + 2*x*A(x) + 6*x^2*A(x)^2 + 56*x^3*A(x)^3 + 1820*x^4*A(x)^4 +...+ C(2^n,n)*x^n*A(x)^n +...

PROG

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, log(1+2^m*x*A+x*O(x^n))^m/m!)); polcoeff(A, n)}

CROSSREFS

Cf. A014070, A188194.

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Mar 23 2011

STATUS

approved

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