A233337 E.g.f. satisfies: A(x) = exp( A(x) * Series_Reversion( Integral A(x) dx ) ).

1, 1, 2, 8, 47, 379, 3824, 46920, 673182, 11117409, 206963008, 4300180282, 98438569429, 2466054159708, 67010276640914, 1965685441214595, 61838543150658378, 2079187591693790811, 74327516147121513022, 2818272509696850645165, 112842691746320772778220, 4763786769795179964384856

Offset

0,3

Comments

a(44) is negative. - Vaclav Kotesovec, Aug 12 2021

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..180

Formula

E.g.f. satisfies: A(x) = LambwertW(-y)/(-y) = Sum_{n>=0} (n+1)^(n-1)*y^n/n! where y = Series_Reversion( Integral A(x) dx ).

Example

E.g.f.: A(x) = 1 + x + 2*x^2/2! + 8*x^3/3! + 47*x^4/4! + 379*x^5/5! + 3824*x^6/6! +...
Related expansions.
log(A(x)) = x + x^2/2! + 4*x^3/3! + 21*x^4/4! + 168*x^5/5! + 1630*x^6/6! +...
The series reversion of the Integral A(x) dx equals log(A(x))/A(x):
log(A(x))/A(x) = x - x^2/2! + x^3/3! - 3*x^4/4! + 8*x^5/5! - 57*x^6/6! + 271*x^7/7! +...
A(Integral A(x) dx) = 1 + x + 3^1*x^2/2! + 4^2*x^3/3! + 5^3*x^4/4! + 6^4*x^5/5! +...+ (n+1)^(n-1)*x^n/n! +... = LambertW(-x)/(-x).

Prog

(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=exp(A*serreverse(intformal(A+x*O(x^n))))); n!*polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Sequence in context: A349588 A259905 A329096 * A199136 A181413 A358822

Adjacent sequences: A233334 A233335 A233336 * A233338 A233339 A233340

Keyword

sign

Author

Paul D. Hanna, Dec 07 2013

Status

approved