A249833 E.g.f. satisfies: A(x) = 1 + Integral A(x) + A(x)^2*log(A(x)) dx.

1, 1, 2, 7, 34, 210, 1574, 13866, 140340, 1604284, 20439484, 287152488, 4409695952, 73482586464, 1320533540808, 25456195929232, 523975944225280, 11469534961767408, 266038450202037728, 6518167274358688512, 168209881653024622944, 4560447490191133853536, 129593625015740116555072

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

Cf. A235129.

Sequence in context: A355292 A002720 A234239 * A111539 A337000 A074059

Adjacent sequences: A249830 A249831 A249832 * A249834 A249835 A249836

Keyword

nonn

Author

Paul D. Hanna, Nov 06 2014

Status

approved