A340450 E.g.f. A(x) satisfies: A(x) = Sum_{n>=0} x^n * exp(x^(n+1)*A(x)).

1, 2, 7, 46, 413, 5036, 77137, 1423276, 30854105, 767958832, 21582020861, 676165597724, 23366484667429, 882971185469296, 36221091773679737, 1603064437712963236, 76138734522477332273, 3862934017045567765568, 208508213805864941722357, 11930655030027036640979500

Offset

0,2

Comments

The e.g.f. A(x) of this sequence is motivated by the following identity:

Sum_{n>=0} p^n * exp(q*r^n) = Sum_{n>=0} (q^n/n!) / (1 - p*r^n) ;

here, p = x, q = x*A(x), and r = x.

Links

Table of n, a(n) for n=0..19.

Formula

E.g.f. A(x) satisfies the following relations.

(1) A(x) = Sum_{n>=0} x^n * exp( x^(n+1)*A(x) ).

(2) A(x) = Sum_{n>=0} (x^n/n!) * A(x)^n / (1 - x^(n+1)).

Example

E.g.f.: A(x) = 1 + 2*x + 7*x^2/2! + 46*x^3/3! + 413*x^4/4! + 5036*x^5/5! + 77137*x^6/6! + 1423276*x^7/7! + 30854105*x^8/8! + 767958832*x^9/9! + ...
where
A(x) = exp(x*A(x)) + x*exp(x^2*A(x)) + x^2*exp(x^3*A(x)) + x^3*exp(x^4*A(x)) + x^4*exp(x^5*A(x)) + x^5*exp(x^6*A(x)) + ...
also
A(x) = 1/(1 - x) + x*A(x)/(1 - x^2) + (x^2/2!)*A(x)^2/(1 - x^3) + (x^3/3!)*A(x)^3/(1 - x^4) + (x^4/4!)*A(x)^4/(1 - x^5) + ...

Prog

(PARI) {a(n) = my(A=1); for(i=1, n, A = sum(m=0, n, x^m * exp(x^(m+1)*A +x*O(x^n)) )); n!*polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n) = my(A=1); for(i=1, n, A = sum(m=0, n, (x^m/m!)*A^m / (1 - x^(m+1) +x*O(x^n)) )); n!*polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A340357.

Sequence in context: A089006 A019004 A027328 * A351917 A349640 A111842

Adjacent sequences: A340447 A340448 A340449 * A340451 A340452 A340453

Keyword

nonn

Author

Paul D. Hanna, Jan 07 2021

Status

approved