|
| |
|
|
A000763
|
|
Number of interval orders constructed from n intervals of generic lengths.
|
|
1
|
|
|
|
1, 3, 19, 195, 2831, 53703, 1264467, 35661979, 1173865927, 44218244943, 1877050837355, 88693432799667, 4618194424504623, 262771389992099719, 16223185411792992403, 1080238361814167993739, 77171781603974127429527
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,2
|
|
|
LINKS
|
|
|
|
FORMULA
|
E.g.f. E(x) satisfies E'/E = y^2, where y=1+x+5*x^2/2+... is defined by y*(2-exp(x*y))=1.
E.g.f.: exp(int(RootOf(2*_Z-_Z*exp(x*_Z)-1)^2, x)) [in Maple notation].
a(n) ~ c * n^(n-2) / (r^n * exp(n)), where r = 2*(LambertW(2*exp(1))-1)^2 / LambertW(2*exp(1)) = 0.204378273928311464700648197201... and c = 1/((1 - 1/LambertW(2*exp(1))) * exp(1/2)*sqrt(2*(1 + 1/LambertW(2*exp(1))))) = 1.196923669815370203369255598062684... . - Vaclav Kotesovec, Mar 22 2016
|
|
|
MAPLE
|
seq(n! * coeff(series(exp(int(RootOf(2*_Z-_Z*exp(x*_Z)-1)^2, x)), x, n+1), x, n), n = 1..20); # Vaclav Kotesovec, Mar 21 2016
|
|
|
MATHEMATICA
|
A000763[max_] := ( e[x_] := Sum[c[k]*x^k, {k, 0, max}]; c[0] = 1; c[1] = 1; y[x_] := Sum[d[k]*x^k, {k, 0, max}]; d[0] = 1; d[1] = 1; cc = CoefficientList[ Series[ e'[x]/e[x] - y[x]^2, {x, 0, max}], x]; dd = CoefficientList[ Series[ y[x]*(2 - Exp[x*y[x]]) - 1, {x, 0, max}], x]; eqdd = Thread[dd == 0]; soldd = Solve[ Thread[dd == 0] ]; eqcc = Thread[(cc /. soldd[[1]]) == 0]; solcc = Solve[ Most[eqcc] ] ; solcc /. Rule -> Set; soldd /. Rule -> Set; Table[c[k], {k, 1, max}] *Range[max]! ); Do[A000763[max], {max, 5, 40, 5}]; A000763[40] (* Jean-François Alcover, Jul 23 2013 *)
|
|
|
PROG
|
(PARI) seq(n)={my(p=serreverse(2*x - x*exp(x + O(x^n)))/x); Vec(serlaplace(exp( intformal(p^2) )))} \\ Andrew Howroyd, Jun 05 2021
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn,nice,easy
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
|
|
|
STATUS
|
approved
|
| |
|
|
