A144012 -id:A144012

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A144013

E.g.f. satisfies: A'(x) = 1 + x*A(x)^3 where A(0) = 1.

+10
3

1, 1, 1, 6, 27, 168, 1395, 12744, 136521, 1672704, 22659993, 340472160, 5605239123, 100154133504, 1933748430939, 40097756280960, 888588043228305, 20962928129900544, 524479633529596209, 13871161113800394240

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..286

FORMULA

E.g.f. satisfies: A(x) = 1 + Integral [1 + x*A(x)^3] dx.

a(n) ~ n^n / (exp(n) * r^(n+1)), where r = 0.69974580613381637... - Vaclav Kotesovec, Feb 24 2014

EXAMPLE

E.g.f.: A(x) = 1 + x + x^2/2! + 6*x^3/3! + 27*x^4/4! + 168*x^5/5! +...

A(x)^3 = 1 + 3*x + 9*x^2/2! + 42*x^3/3! + 279*x^4/4! + 2124*x^5/5! +...

x*A(x)^3 = x + 6*x^2/2! + 27*x^3/3! + 168*x^4/4! + 1395*x^5/5! +...

A'(x) = 1 + x + 6*x^2/2! + 27*x^3/3! + 168*x^4/4! + 1395*x^5/5! +...

PROG

(PARI) {a(n)=local(A=1+x); for(i=0, n, A=1+intformal(1+x*(A+x*O(x^n))^3)); n!*polcoeff(A, n)}

CROSSREFS

Cf. A144012, A144014.

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Sep 10 2008

STATUS

approved

A144014

E.g.f. satisfies: A'(x) = 1 + x*A(x)^4 where A(0) = 1.

+10
3

1, 1, 1, 8, 48, 368, 3900, 46992, 647472, 10294848, 182582424, 3576144000, 76958290464, 1801577086848, 45572841133248, 1239448991058432, 36058780518552000, 1117339391835583488, 36741513671695717632

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..312

FORMULA

E.g.f. satisfies: A(x) = 1 + Integral [1 + x*A(x)^4] dx.

a(n) ~ n^(n-1/6) * sqrt(2*Pi) / (3^(1/3) * GAMMA(1/3) * exp(n) * r^(n+2/3)), where r = 0.52731343741213... (multiplicative constant is conjectured, holds 14 decimal places). - Vaclav Kotesovec, Feb 24 2014

EXAMPLE

E.g.f.: A(x) = 1 + x + x^2/2! + 8*x^3/3! + 48*x^4/4! + 368*x^5/5! +...

A(x)^4 = 1 + 4*x + 16*x^2/2! + 92*x^3/3! + 780*x^4/4! + 7832*x^5/5! +...

x*A(x)^4 = x + 8*x^2/2! + 48*x^3/3! + 368*x^4/4! + 3900*x^5/5! +...

A'(x) = 1 + x + 8*x^2/2! + 48*x^3/3! + 368*x^4/4! + 3900*x^5/5! +...