A266328 E.g.f. A(x) satisfies: A(x) = exp( Integral B(x) dx ) such that B(x) = exp(-x) * exp( Integral A(x) dx ), where the constant of integration is zero.

1, 1, 1, 2, 6, 21, 92, 469, 2731, 17985, 131528, 1059616, 9319363, 88833422, 912393381, 10043727089, 117969438513, 1472593659884, 19467505081458, 271704942613323, 3992343851680466, 61603531051030691, 995949139457447931, 16835191741257445589, 296976010796327785530, 5457427389713208932740, 104308245862443706265341, 2070461793105333579698992, 42622090166454492404075635

Offset

0,4

Comments

Compare to: G(x) = exp( Integral G(x) dx ) when G(x) = 1/(1-x).

What is lim_{n->oo} (a(n)/n!)^(1/n)? Example: (a(500)/500!)^(1/500) = 0.7353325805...

Limit_{n->oo} (a(n)/n!)^(1/n) = 1/Integral_{x=0..oo} 1/(exp(x) - x) dx = 0.73578196429164719984313538... - Vaclav Kotesovec, Aug 21 2017

Links

Paul D. Hanna, Table of n, a(n) for n = 0..300

Formula

E.g.f. A(x) satisfies:

(1) A(x) = exp( Integral A(x) - log(A(x)) dx ).

(2) A(x) = log(A(x)) + A'(x)/A(x).

(3) log(A(x)) = exp(-x) * Integral exp(x)*A(x) dx.

(4) A(x) = exp( Series_Reversion( Integral 1/(exp(x) - x) dx ) ).

a(n) ~ c^(n+1) * n!, where c = 1/Integral_{x=0..oo} 1/(exp(x) - x) dx = 0.7357819642916471998431353808137704665788888148929882090175... - Vaclav Kotesovec, Aug 21 2017

Conjecture: a(n) = R(n-1, 0) for n > 0 with a(0) = 1 where R(n, q) = R(n-1, q+1) + Sum_{j=0..q-1} binomial(q+1, j)*R(n-1, j) for n > 0, q >= 0 with R(0, q) = 1 for q >= 0. - Mikhail Kurkov, Dec 26 2023

Example

E.g.f.: A(x) = 1 + x + x^2/2! + 2*x^3/3! + 6*x^4/4! + 21*x^5/5! + 92*x^6/6! + 469*x^7/7! + 2731*x^8/8! + 17985*x^9/9! + 131528*x^10/10! + ...
such that log(A(x)) = Integral B(x) dx
where
B(x) = 1 + x^2/2! + x^3/3! + 5*x^4/4! + 16*x^5/5! + 76*x^6/6! + 393*x^7/7! + 2338*x^8/8! + 15647*x^9/9! + 115881*x^10/10! + ...
and A(x) and B(x) satisfy:
(1) A(x) = B'(x)/B(x) + 1,
(2) B(x) = A'(x)/A(x),
(3) B(x) = A(x) - log(A(x)),
(4) log(A(x)) = Integral B(x) dx,
(5) log(B(x)) = Integral A(x) dx - x.
The Series Reversion of log(A(x)) equals Integral 1/(exp(x) - x) dx:
Integral 1/(exp(x) - x) dx  =  x - x^3/3! - x^4/4! + 5*x^5/5! + 19*x^6/6! - 41*x^7/7! - 519*x^8/8! - 183*x^9/9! + 19223*x^10/10! + ... + A089148(n-1)*x^n/n! + ...
so that A( Integral 1/(exp(x) - x) dx ) = exp(x).

Prog

(PARI) {a(n) = my(A=1+x, B=1+x); for(i=0, n, A = exp( intformal( B + x*O(x^n) ) ); B = exp( intformal( A - 1 ) ) ); n!*polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n) = n! * polcoeff( exp( serreverse( intformal( 1/(exp(x +x*O(x^n)) - x) ) )), n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) Vec( serlaplace( exp( serreverse( intformal( 1/(exp(x +x*O(x^25)) - x)))))) \\ Joerg Arndt, Dec 26 2023

Crossrefs

Cf. A006789, A089148, A266329.

Sequence in context: A231166 A115089 A304196 * A214087 A183950 A001928

Adjacent sequences: A266325 A266326 A266327 * A266329 A266330 A266331

Keyword

nonn

Author

Paul D. Hanna, Jan 24 2016

Status

approved