|
| |
|
|
A193363
|
|
O.g.f. satisfies: A(x) = Sum_{n>=0} (n+1)^n * x^n * A((n+1)*x)^n/n! * exp(-(n+1)*x*A((n+1)*x)).
|
|
8
|
|
|
|
1, 1, 4, 41, 871, 36137, 2885457, 443469511, 131707909982, 75945551138638, 85425571722359386, 188277619627892581987, 816318863956958720950775, 6986374103851011507327849798, 118360360643974268213872443877649, 3978536338453184605328853807076468581
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
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)).
|
|
|
LINKS
|
|
|
|
EXAMPLE
|
O.g.f.: A(x) = 1 + x + 4*x^2 + 41*x^3 + 871*x^4 + 36137*x^5 + 2885457*x^6 +...
where
A(x) = exp(-x*A(x)) + 2*x*A(2*x)*exp(-2*x*A(2*x)) + 3^2*x^2*A(3*x)^2/2!*exp(-3*x*A(3*x)) + 4^3*x^3*A(4*x)^3/3!*exp(-4*x*A(4*x)) + 5^4*x^4*A(5*x)^4/4!*exp(-5*x*A(5*x)) + 6^5*x^5*A(6*x)^5/5!*exp(-6*x*A(6*x)) +...
simplifies to a power series in x with integer coefficients.
|
|
|
MATHEMATICA
|
A[_] = 0; m = 16;
Do[A[x_] = Exp[-x A[x]] + Sum[(n+1)^n x^n A[(n+1)x]^n/n! Exp[-(n+1) x A[(n+1)x]], {n, 1, m}] + O[x]^m // Normal, {m}];
|
|
|
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+1)*x)^k/k!*exp(-(k+1)*x*subst(A, x, (k+1)*x)+x*O(x^n)))); polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
