login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A268170 E.g.f. A(x) satisfies: A(x) = exp( Integral B(x) dx ) such that B(x) = exp(1+x - exp(x)) * exp( Integral A(x) dx ), where the constant of integration is zero. 1
1, 1, 2, 5, 16, 65, 326, 1947, 13410, 104181, 900214, 8566655, 89055224, 1004141647, 12204369138, 159036267519, 2211764983734, 32696763676521, 511987792322430, 8465194670035767, 147370831072230860, 2694506417687396995, 51622643862824956898, 1034153511794063402519, 21621325640846679627146 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Compare to: F(x) = exp( Integral G(x) dx ) such that G(x) = exp(1-exp(x)) * exp( Integral F(x) dx ) holds when F(x) = exp(x).

LINKS

Table of n, a(n) for n=0..24.

EXAMPLE

E.g.f.: A(x) = 1 + x + 2*x^2/2! + 5*x^3/3! + 16*x^4/4! + 65*x^5/5! + 326*x^6/6! + 1947*x^7/7! + 13410*x^8/8! + 104181*x^9/9! + 900214*x^10/10! + 8566655*x^11/11! +...

such that log(A(x)) = Integral B(x) dx

where

B(x) = 1 + x + x^2/2! + 2*x^3/3! + 9*x^4/4! + 46*x^5/5! + 245*x^6/6! + 1474*x^7/7! + 10315*x^8/8! + 82174*x^9/9! + 726591*x^10/10! + 7038632*x^11/11! + 74216949*x^12/12! +...+ A268171(n)*x^n/n! +...

and A(x) and B(x) satisfy:

(1) A(x) = B'(x)/B(x) + exp(x) - 1,

(2) B(x) = A'(x)/A(x),

(3) log(A(x)) = Integral B(x) dx,

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

RELATED SERIES.

log(A(x)) = x + x^2/2! + x^3/3! + 2*x^4/4! + 9*x^5/5! + 46*x^6/6! + 245*x^7/7! + 1474*x^8/8! + 10315*x^9/9! + 82174*x^10/10! + 726591*x^11/11! + 7038632*x^12/12! +...

Let J(x) equal the series reversion of log(A(x)); then

J(x) = x - x^2/2! + 2*x^3/3! - 7*x^4/4! + 31*x^5/5! - 172*x^6/6! + 1155*x^7/7! - 9027*x^8/8! + 80676*x^9/9! - 811727*x^10/10! + 9075333*x^11/11! - 111633356*x^12/12! +...

where A(J(x)) = 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(1+x - exp(x +x*O(x^n)) + intformal( A ) ) ); n!*polcoeff(A, n)}

for(n=0, 30, print1(a(n), ", "))

CROSSREFS

Cf. A266328, A266329, A266490, A268171.

Sequence in context: A131178 A003149 A027046 * A000522 A182290 A007469

Adjacent sequences:  A268167 A268168 A268169 * A268171 A268172 A268173

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jan 27 2016

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified September 24 14:46 EDT 2021. Contains 347643 sequences. (Running on oeis4.)