|
| |
|
|
A368630
|
|
Expansion of e.g.f. A(x) satisfying A(x/A(x)^2) = exp(x*A(x)).
|
|
2
|
|
|
|
1, 1, 7, 136, 4933, 275536, 21309139, 2137447936, 266227499017, 39924910381312, 7045914488563711, 1437809941831499776, 334581893955246072205, 87792555944973238718464, 25735892905876612366925515, 8363132129019712402301648896, 2992768723058093966270081891089
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,3
|
|
|
COMMENTS
|
Conjecture: a(n) == 1 (mod 3) for n >= 0.
Conjecture: a(2*n) == 1 (mod 2) for n >= 0.
Conjecture: a(2*n+1) == 0 (mod 2) for n >= 1.
|
|
|
LINKS
|
|
|
|
FORMULA
|
E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.
(1) A(x/A(x)^2) = exp(x*A(x)).
(2) A(x) = exp(x*B(x)^3) where B(x) = A(x*B(x)^2) = ( (1/x)*Series_Reversion(x/A(x)^2) )^(1/2).
(3) A(x/C(x)^3) = exp(x) where C(x) = A(x/C(x)) = x/Series_Reversion(x*A(x)).
|
|
|
EXAMPLE
|
E.g.f.: A(x) = 1 + x + 7*x^2/2! + 136*x^3/3! + 4933*x^4/4! + 275536*x^5/5! + 21309139*x^6/6! + 2137447936*x^7/7! + 266227499017*x^8/8! + ...
where A(x/A(x)^2) = exp(x*A(x)) and
exp(x*A(x)) = 1 + x + 3*x^2/2! + 28*x^3/3! + 653*x^4/4! + 28096*x^5/5! + 1833367*x^6/6! + 162874048*x^7/7! + ...
Also,
A(x) = exp(x*B(x)^3) where B(x) = A(x*B(x)^2) begins
B(x) = 1 + x + 11*x^2/2! + 292*x^3/3! + 13149*x^4/4! + 861376*x^5/5! + 75412591*x^6/6! + 8365301568*x^7/7! + ...
B(x)^2 = 1 + 2*x + 24*x^2/2! + 650*x^3/3! + 29360*x^4/4! + 1918482*x^5/5! + 167206144*x^6/6! + ...
B(x)^3 = 1 + 3*x + 39*x^2/2! + 1080*x^3/3! + 49029*x^4/4! + 3199728*x^5/5! + 277840179*x^6/6! + ...
Further,
A(x/C(x)^3) = exp(x) where C(x) = A(x/C(x)) begins
C(x) = 1 + x + 5*x^2/2! + 85*x^3/3! + 2889*x^4/4! + 154441*x^5/5! + 11527693*x^6/6! + 1120674717*x^7/7! + ...
C(x)^3 = 1 + 3*x + 21*x^2/2! + 351*x^3/3! + 11337*x^4/4! + 582843*x^5/5! + 42300765*x^6/6! + ...
|
|
|
PROG
|
(PARI) {a(n) = my(A=1+x); for(i=0, n, A = exp( x*((1/x)*serreverse( x/(A^2 + x*O(x^n)) ))^(3/2) )); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
