A233336 -id:A233336 - cp's OEIS frontend

A233335 E.g.f. A(x) satisfies: A( Integral 1/A(x) dx ) = exp(x).

1, 1, 2, 7, 38, 292, 2975, 38350, 604433, 11351659, 249042701, 6283114723, 179995680530, 5794486077958, 207806806310354, 8241414107222095, 359171801820266717, 17107537203463252273, 886296777786378900077, 49732564234138336160086, 3011177123882906437153214, 196063383282648338166793297

Offset: 0

Paul D. Hanna, Dec 07 2013

nonn

			E.g.f.: A(x) = 1 + x + 2*x^2/2! + 7*x^3/3! + 38*x^4/4! + 292*x^5/5! + 2975*x^6/6! +...
Related expansions.
Integral 1/A(x) dx = x - x^2/2! - x^4/4! - 6*x^5/5! - 52*x^6/6! - 591*x^7/7! - 8404*x^8/8! +...
The series reversion of Integral 1/A(x) dx equals log(A(x)) and begins:
log(A(x)) = x + x^2/2! + 3*x^3/3! + 16*x^4/4! + 126*x^5/5! + 1333*x^6/6! + 17895*x^7/7! + 293461*x^8/8! +...+ A214645(n)*x^n/n! +...
and equals the e.g.f. of A214645.

Paul D. Hanna, Table of n, a(n) for n = 0..160

Cf. A233336, A214645 (log).

{a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=exp(serreverse(intformal(1/A+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) dx ) ).

E.g.f.: exp(G(x)) where G(x) = exp(G(G(x))) is the e.g.f. of A214645.

A259267 E.g.f. A(x) satisfies: A'(x) = exp(2*A(A(x))).

Original entry on oeis.org

1, 2, 12, 128, 2016, 42656, 1145280, 37563008, 1464675840, 66533778944, 3466031815680, 204489094565888, 13524452573872128, 994257291909816320, 80668058806271016960, 7179145234347383128064, 697131195162680465817600, 73522035747248454761578496, 8387016414085244676889116672

Offset: 1

Paul D. Hanna, Jul 03 2015

nonn

			E.g.f.: A(x) = x + 2*x^2/2! + 12*x^3/3! + 128*x^4/4! + 2016*x^5/5! +...
Related expansions:
A'(x) =  1 + 2*x + 12*x^2/2! + 128*x^3/3! + 2016*x^4/4! + 42656*x^5/5! +...
A(A(x)) = log(A'(x))/2 = x + 4*x^2/2! + 36*x^3/3! + 520*x^4/4! + 10512*x^5/5! + 276064*x^6/6! + 8987712*x^7/7! + 351278080*x^8/8! +...
The exponential of e.g.f. A(x) equals the e.g.f. of A233336:
exp(A(x)) = 1 + x + 3*x^2/2! + 19*x^3/3! + 201*x^4/4! + 3097*x^5/5! + 63963*x^6/6! + 1677883*x^7/7! +...+ A233336(n)*x^n/n! +...

Cf. A233336, A214645.

{a(n)=local(A=x+x^2); for(i=0, n, A=intformal(exp(2*subst(A, x, A+x*O(x^n))))); n!*polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))

E.g.f. A(x) satisfies:
(1) A''(x) = 2*exp( 4*A(A(x)) + 2*A(A(A(x))) ).
(2) exp(-2*A(x)) = d/dx Series_Reversion(A(x)).
(3) A(x) = log(F(x)) where F(x) satisfies: F( Integral 1/F(x)^2 dx ) = exp(x) and equals the e.g.f. of A233336.
a(n) = 2^(n-1) * A214645(n) for n>=1.

cp's OEIS Frontend

A233335 E.g.f. A(x) satisfies: A( Integral 1/A(x) dx ) = exp(x).

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