STATUS
editing
approved
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O.g.f. satisfies: A(x) = Sum_{n>=0} (n+1)^n * x^n * A(n*x)^n/n! * exp(-(n+1)*x*A(n*x)).
(history; published version)
(history; published version)
STATUS
editing
approved
STATUS
editing
approved
STATUS
editing
approved
NAME
O.g.f. satisfies: A(x) = Sum_{n>=0} (n+1)^n * x^n * A(n*x)^n/n! * exp(-(n+1)*x*A(n*x)).
COMMENTS
Compare to the LambertW identity:
EXAMPLE
O.g.f.: A(x) = 1 + x + 3*x^2 + 17*x^3 + 160*x^4 + 2209*x^5 + 44081*x^6 +...
PROG
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(k=0, n, (k+1)^k*x^k*subst(A, x, k*x)^k/k!*exp(-(k+1)*x*subst(A, x, k*x)+x*O(x^n)))); polcoeff(A, n)}
NAME
allocated for Paul D. Hanna
O.g.f. satisfies: A(x) = Sum_{n>=0} (n+1)^n * x^n * A(n*x)^n/n! * exp(-(n+1)*A(n*x)).
DATA
1, 1, 3, 17, 160, 2209, 44081, 1247278, 50003383, 2843143785, 229717311597, 26423288336013, 4331881870569310, 1013060852125392519, 338180578288458076194, 161225876602752196310870, 109821236456762132613619651, 106923122485613725232770276036
OFFSET
0,3
COMMENTS
Compare to the LambertW identity:
Sum_{n>=0} (n+1)^n * x^n * G(x)^n/n! * exp(-(n+1)*x*G(x)) = 1/(1 - x*G(x)).
EXAMPLE
O.g.f.: A(x) = 1 + x + 3*x^2 + 17*x^3 + 160*x^4 + 2209*x^5 + 44081*x^6 +...
where
A(x) = exp(-x) + 2*x*A(x)*exp(-2*x*A(x)) + 3^2*x^2*A(2*x)^2/2!*exp(-3*x*A(2*x)) + 4^3*x^3*A(3*x)^3/3!*exp(-4*x*A(3*x)) + 5^4*x^4*A(4*x)^4/4!*exp(-5*x*A(4*x)) + 6^5*x^5*A(5*x)^5/5!*exp(-6*x*A(5*x)) +...
simplifies to a power series in x with integer coefficients.
PROG
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(k=0, n, (k+1)^k*x^k*subst(A, x, k*x)^k/k!*exp(-(k+1)*x*subst(A, x, k*x)+x*O(x^n)))); polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
KEYWORD
allocated
nonn
AUTHOR
Paul D. Hanna, Jan 15 2013
STATUS
approved
editing
NAME
allocated for Paul D. Hanna
KEYWORD
allocated
STATUS
approved