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)).

1, 1, 4, 41, 871, 36137, 2885457, 443469511, 131707909982, 75945551138638, 85425571722359386, 188277619627892581987, 816318863956958720950775, 6986374103851011507327849798, 118360360643974268213872443877649, 3978536338453184605328853807076468581

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

Table of n, a(n) for n=0..15.

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}];
CoefficientList[A[x], x] (* Jean-François Alcover, Oct 29 2019 *)

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

Cf. A218672, A209276, A209277.

Cf. A221409, A221410, A221411, A221412, A221413.

Sequence in context: A006129 A244437 A265003 * A284276 A327026 A244751

Adjacent sequences: A193360 A193361 A193362 * A193364 A193365 A193366

Keyword

nonn

Author

Paul D. Hanna, Jan 09 2013

Status

approved