OFFSET
1,2
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..100
FORMULA
E.g.f. A(x) = Sum_{n>=1} a(n) * x^n / n! satisfies:
(1) A( A( x^5*exp(5*x) )^(1/5) ) = -LambertW(-x*exp(x)).
(2) A(x) = Series_Reversion( A( x^5*exp(-5*x) )^(1/5) ).
(3) A( A(x)^5 * exp(-5*A(x)) ) = x^5.
EXAMPLE
E.g.f.: A(x) = x + 2*x^2/2! + 9*x^3/3! + 64*x^4/4! + 625*x^5/5! + 7632*x^6/6! + 115633*x^7/7! + 2060864*x^8/8! + 42272577*x^9/9! + 981100000*x^10/10! + 25420901209*x^11/11! + 727392785472*x^12/12! + 22781551770289*x^13/13! + 775174385740496*x^14/14! + 28475611427390625*x^15/15! +...
such that A( A( x^5*exp(-5*x) )^(1/5) ) = x.
RELATED SERIES.
The series reversion of the e.g.f. A(x) equals the series defined by:
A( x^5*exp(-5*x) )^(1/5) = x - 2*x^2/2! + 3*x^3/3! - 4*x^4/4! + 5*x^5/5! + 138*x^6/6! - 6041*x^7/7! + 145144*x^8/8! - 2612727*x^9/9! + 39191030*x^10/10! - 508540021*x^11/11! + 5048676852*x^12/12! + 13708341517*x^13/13! - 3528271498766*x^14/14! + 168238690139535*x^15/15! +...
Compare the above series reversion to the following series:
A(x)^5 * exp(-5*A(x)) = x^5 - 2*x^10/2! + 3*x^15/3! - 4*x^20/4! + 5*x^25/5! + 138*x^30/6! - 6041*x^35/7! + 145144*x^40/8! - 2612727*x^45/9! +...
where A( A(x)^5 * exp(-5*A(x)) ) = x^5.
Note the following series is also in powers of x^5:
A(-A(x)^5 * exp(-5*A(x)) ) = -x^5 + 4*x^10/2! - 24*x^15/3! + 224*x^20/4! - 2880*x^25/5! + 46944*x^30/6! - 926464*x^35/7! + 21582976*x^40/8! - 581587200*x^45/9! + 17791870400*x^50/10! - 608466076416*x^55/11! + 22988808466560*x^60/12! - 950707246757632*x^65/13! + 42712071655886752*x^70/14! - 2071447871355102720*x^75/15! + 107861157493089761024*x^80/16! +...
PROG
(PARI) {a(n) = my(A=x); for(i=1, n, A = serreverse( subst(A, x, x^5*exp(-5*x +x*O(x^n)))^(1/5) ) ); n!*polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 21 2016
STATUS
approved