|
| |
|
|
A274395
|
|
E.g.f. A(x) satisfies: A( A( x^5*exp(-5*x) )^(1/5) ) = x.
|
|
4
|
|
|
|
1, 2, 9, 64, 625, 7632, 115633, 2060864, 42272577, 981100000, 25420901209, 727392785472, 22781551770289, 775174385740496, 28475611427390625, 1123174379270470528, 47345176946662808833, 2124056646149570472384, 101049649535116764217513, 5081280208216339430000000, 269289663191356712678537841, 15001629187601225176466619952, 876397229390129697388339920049
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,2
|
|
|
LINKS
|
|
|
|
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
|
|
|
|
STATUS
|
approved
|
| |
|
|
