OFFSET
0,3
COMMENTS
Compare to e.g.f. of Bell numbers: if B(x) = exp(exp(x)-1) then
B(x) = 1 + Integral B(x) + B(x)*log(B(x)) dx.
Limit_{n->oo} (a(n)/n!)^(1/n) = 1.30339... (cf. A235129). - Vaclav Kotesovec, Nov 09 2014
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..230
FORMULA
E.g.f. satisfies: A(x) = exp( Integral 1 + A(x)*log(A(x)) dx ).
Conjecture: a(n) = A370382(n-1, 0) for n > 0 with a(0) = 1. - Mikhail Kurkov, Apr 25 2024
EXAMPLE
E.g.f.: A(x) = 1 + x + 2*x^2/2! + 7*x^3/3! + 34*x^4/4! + 210*x^5/5! + ...
Related expansions.
A(x)^2*log(A(x)) = x + 5*x^2/2! + 27*x^3/3! + 176*x^4/4! + 1364*x^5/5! + ...
A(x)^2 = 1 + 2*x + 6*x^2/2! + 26*x^3/3! + 148*x^4/4! + 1040*x^5/5! + 8688*x^6/6! + 84068*x^7/7! + 924384*x^8/8! + 11381696*x^9/9! + ...
log(A(x)) = x + x^2/2! + 3*x^3/3! + 12*x^4/4! + 64*x^5/5! + 424*x^6/6! + 3358*x^7/7! + 30952*x^8/8! + 325488*x^9/9! + 3845724*x^10/10! + ...
PROG
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+intformal(A+A^2*log(A +x*O(x^n)))); n!*polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=exp(intformal(1+A*log(A +x*O(x^n))))); n!*polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Nov 06 2014
STATUS
approved