|
| |
|
|
A276908
|
|
E.g.f. A(x) satisfies: Series_Reversion( LambertW(A(x)) ) = -LambertW(-A(x)).
|
|
3
|
|
|
|
1, 0, -3, 0, 5, 0, 609, 0, -49239, 0, 13360325, 0, -15252271827, 0, 37424782201065, 0, -170209716095463727, 0, 1318010670452522053773, 0, -16258330635018856956323115, 0, 303578009078402791805745874289, 0, -8239767220259502118041112282583175, 0, 314540395737200378455353074387646404949, 0, -16429342682031325194203779796586447506765059, 0, 1147197863673453312133563419500631514117221228025, 0
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,3
|
|
|
COMMENTS
|
It appears that a(6*k+5) = 2 (mod 3) for k>=0 with a(n) = 0 (mod 3) elsewhere.
E.g.f. A(x) equals the series reversion of the e.g.f. of A276909.
|
|
|
LINKS
|
|
|
|
FORMULA
|
E.g.f. A(x) satisfies:
(1) A( LambertW(A(x)) ) = x*exp(-x),
(2) A( -LambertW(-A(x)) ) = x*exp(x),
where LambertW( x*exp(x) ) = x.
(3) Series_Reversion( A( LambertW(x) ) ) = A( -LambertW(-x) ).
|
|
|
EXAMPLE
|
E.g.f.: A(x) = x - 3*x^3/3! + 5*x^5/5! + 609*x^7/7! - 49239*x^9/9! + 13360325*x^11/11! - 15252271827*x^13/13! + 37424782201065*x^15/15! - 170209716095463727*x^17/17! + 1318010670452522053773*x^19/19! - 16258330635018856956323115*x^21/21! + 303578009078402791805745874289*x^23/23! - 8239767220259502118041112282583175*x^25/25! +...
such that Series_Reversion( LambertW(A(x)) ) = LambertW(-A(x)).
RELATED SERIES.
LambertW(A(x)) = x - 2*x^2/2! + 6*x^3/3! - 40*x^4/4! + 360*x^5/5! - 4176*x^6/6! + 59248*x^7/7! - 978048*x^8/8! + 18529920*x^9/9! - 397792000*x^10/10! + 9541483776*x^11/11! - 252320449536*x^12/12! + 7287410271232*x^13/13! - 229140322854912*x^14/14! + 7813274134640640*x^15/15! - 284528460591824896*x^16/16! + 10907160860365848576*x^17/17! - 452737238590418780160*x^18/18! + 21213627569654340321280*x^19/19! +...
exp(A(x)) = 1 + x + x^2/2! - 2*x^3/3! - 11*x^4/4! - 24*x^5/5! + 61*x^6/6! + 1240*x^7/7! + 6665*x^8/8! - 34496*x^9/9! - 728999*x^10/10! + 8173056*x^11/11! + 172370749*x^12/12! - 13734849920*x^13/13! - 218875559083*x^14/14! + 35623302896896*x^15/15! + 613795913727121*x^16/16! +...
Also, A( LambertW(A(x)) ) = x*exp(-x), where
LambertW(x) = x - 2*x^2/2! + 9*x^3/3! - 64*x^4/4! + 625*x^5/5! - 7776*x^6/6! + 117649*x^7/7! - 2097152*x^8/8! +...+ -n^(n-1)*(-x)^n/n! +...
|
|
|
PROG
|
(PARI) {a(n) = my(A=x +x*O(x^n), W = serreverse(x*exp(x +x*O(x^n))));
for(i=1, n, A = A + (x - subst(subst(W, x, A), x, -subst(W, x, -A)))/2); n!*polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
sign
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
