A233335 -id:A233335 - cp's OEIS frontend

A214645 E.g.f. A(x) satisfies: A'(x) = exp(A(A(x))).

1, 1, 3, 16, 126, 1333, 17895, 293461, 5721390, 129948787, 3384796695, 99848190706, 3301868304168, 121369298328835, 4923587573624940, 219090125559917698, 10637377855875861600, 560928617456424367993, 31993928581562975604588, 1966682218962058310721178

Offset: 1

Paul D. Hanna, Jul 24 2012

nonn

The e.g.f A(x) is divergent according to the answer by Pietro Majer to the MathOverflow question linked below. - Tom Copeland, Jan 16 2017

			E.g.f.: A(x) = x + x^2/2! + 3*x^3/3! + 16*x^4/4! + 126*x^5/5! + 1333*x^6/6! + ...
Related expansions:
A'(x) = 1 + x + 3*x^2/2! + 16*x^3/3! + 126*x^4/4! + 1333*x^5/5! + ...
A(A(x)) = log(A'(x)) = x + 2*x^2/2! + 9*x^3/3! + 65*x^4/4! + 657*x^5/5! + 8627*x^6/6! + 140433*x^7/7! + 2744360*x^8/8! + 62894577*x^9/9! + ...
The exponential of e.g.f. A(x) equals the e.g.f. of A233335:
exp(A(x)) = 1 + x + 2*x^2/2! + 7*x^3/3! + 38*x^4/4! + 292*x^5/5! + 2975*x^6/6! + 38350*x^7/7! + 604433*x^8/8! + 11351659*x^9/9! + ... + A233335(n)*x^n/n! + ...

Paul D. Hanna, Table of n, a(n) for n = 1..160
MathOverflow, How do I solve this: df/dx = exp[f^{(-1)}(x)], answer by Tom Copeland to a MathOverflow question by Zeraoulia Rafik, 2017.
MathOverflow, df/dx = exp[f^{(-1)}(x)] again, answer by Pietro Majer to a MathOverflow question by Fan Zheng, 2017.

Cf. A233335 (exp), A259267.

{a(n)=local(A=x+x^2);for(i=0,n,A=intformal(exp(subst(A,x,A+x*O(x^n)))));n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

E.g.f. A(x) satisfies:
(1) A''(x) = exp( 2*A(A(x)) + A(A(A(x))) ).
(2) exp(-A(x)) = d/dx Series_Reversion(A(x)).
(3) A(x) = Series_Reversion( Integral exp(-A(x)) dx ).
(4) A(x) = log(F(x)) where F(x) satisfies F( Integral 1/F(x) dx ) = exp(x) and equals the e.g.f. of A233335.

A233336 E.g.f. A(x) satisfies: A( Integral 1/A(x)^2 dx ) = exp(x).

Original entry on oeis.org

1, 1, 3, 19, 201, 3097, 63963, 1677883, 53862225, 2059533745, 91909156083, 4711414480867, 273922577628057, 17876478783834313, 1298278981158825291, 104165674946626478347, 9175884838706696138145, 882669439812976183138657, 92284482487864563215652579, 10443186091555501868233274803

Offset: 0

Paul D. Hanna, Dec 07 2013

nonn

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 19*x^3/3! + 201*x^4/4! + 3097*x^5/5! + 63963*x^6/6! +...
Related expansions.
A(x)^2 = 1 + 2*x + 8*x^2/2! + 56*x^3/3! + 608*x^4/4! + 9344*x^5/5! + 190400*x^6/6! +...+ A233335(n)*(2*x)^n/n! +...
Integral 1/A(x)^2 dx = x - 2*x^2/2! - 8*x^4/4! - 96*x^5/5! - 1664*x^6/6! +...
The series reversion of the Integral 1/A(x)^2 dx equals log(A(x)) and begins:
log(A(x)) = x + 2*x^2/2! + 12*x^3/3! + 128*x^4/4! + 2016*x^5/5! + 42656*x^6/6! + 1145280*x^7/7! + 37563008*x^8/8! +...+ A259267(n)*x^n/n! +...

Cf. A233335, A259267.

{a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=exp(serreverse(intformal(1/A^2+x*O(x^n)))));n!*polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

E.g.f. satisfies: A(x) = exp( Series_Reversion( Integral 1/A(x)^2 dx ) ).

A259822 E.g.f. A(x) satisfies: A( Integral 1/A(x)^3 dx ) = exp(x).

Original entry on oeis.org

1, 1, 4, 37, 586, 13612, 424621, 16827976, 815866699, 47093387797, 3170897237125, 245127016240321, 21482473673228266, 2112385883734692910, 231062843227493844112, 27913223028923592662539, 3701041353685453743060265, 535729316331363978105167557, 84263588534262286958390813305

Offset: 0

Paul D. Hanna, Jul 05 2015

nonn

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 37*x^3/3! + 586*x^4/4! + 13612*x^5/5! + 424621*x^6/6! +...
where log(A(x)) = Series_Reversion( Integral 1/A(x)^3 dx ):
log(A(x)) = x + 3*x^2/2! + 27*x^3/3! + 432*x^4/4! + 10206*x^5/5! + 323919*x^6/6! +...+ 3^(n-1)*A214645(n)*x^n/n! +...
and
A(x/3)^3 = 1 + x + 2*x^2/2! + 7*x^3/3! + 38*x^4/4! + 292*x^5/5! + 2975*x^6/6! +...+ A233335(n)*x^n/n! +...

Cf. A233335, A214645.

{a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=exp(serreverse(intformal(1/A^3+x*O(x^n))))); n!*polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

E.g.f. satisfies: A(x) = exp( Series_Reversion( Integral 1/A(x)^3 dx ) ).

E.g.f. A(x) such that A(x/3)^3 is the e.g.f. of A233335.

cp's OEIS Frontend

A214645 E.g.f. A(x) satisfies: A'(x) = exp(A(A(x))).

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A233336 E.g.f. A(x) satisfies: A( Integral 1/A(x)^2 dx ) = exp(x).

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A259822 E.g.f. A(x) satisfies: A( Integral 1/A(x)^3 dx ) = exp(x).

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