OFFSET
0,3
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..150
FORMULA
E.g.f. satisfies:
(1) A(x) = exp(x) * (1 + x^2*A'(x)/A(x)).
(2) A(x) = exp( Sum_{n>=1} A245119(n+1)*x^n/n ).
a(n) ~ c * (n!)^2 / n, where c = BesselJ(1,2) = 0.5767248077568733872... . - Vaclav Kotesovec, Jul 22 2014
EXAMPLE
E.g.f.: A(x) = 1 + x + 3*x^2/2! + 19*x^3/3! + 205*x^4/4! + 3501*x^5/5! +...
such that A(x) = exp(x) * (1 + x^2*A'(x)/A(x)) where
1 + x^2*A'(x)/A(x) = 1 + 2*x^2/2! + 12*x^3/3! + 144*x^4/4! + 2640*x^5/5! + 72000*x^6/6! + 2792160*x^7/7! + 147329280*x^8/8! +...
RELATED SERIES.
The e.g.f. equals the product of exp(x) and an integer series (A245119):
exp(-x)*A(x) = 1 + x^2 + 2*x^3 + 6*x^4 + 22*x^5 + 100*x^6 + 554*x^7 + 3654*x^8 + 28014*x^9 + 244572*x^10 + 2392042*x^11 + 25877610*x^12 +...+ A245119(n)*x^n +...
The logarithmic derivative of the e.g.f. is an integer series:
A'(x)/A(x) = 1 + 2*x + 6*x^2 + 22*x^3 + 100*x^4 + 554*x^5 + 3654*x^6 + 28014*x^7 + 244572*x^8 + 2392042*x^9 + 25877610*x^10 +...+ A245119(n+2)*x^n +...
PROG
(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=sum(m=0, n, x^m*deriv(x*A^m)/A^m/m!+x*O(x^n))); n!*polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=sum(m=0, n, x^m*(1+m*x*A'/A)/m!+x*O(x^n))); n!*polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jul 21 2014
STATUS
approved