A000763 - OEIS

%I #55 Jun 05 2021 16:36:48

%S 1,3,19,195,2831,53703,1264467,35661979,1173865927,44218244943,

%T 1877050837355,88693432799667,4618194424504623,262771389992099719,

%U 16223185411792992403,1080238361814167993739,77171781603974127429527

%N Number of interval orders constructed from n intervals of generic lengths.

%H Vincenzo Librandi, Jean-François Alcover and Bruno Berselli, <a href="/A000763/b000763.txt">Table of n, a(n) for n = 1..100</a> (up to n = 21 from _Vincenzo Librandi_, up to n = 40 from _Jean-François Alcover_)

%F 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.

%F E.g.f.: exp(int(RootOf(2*_Z-_Z*exp(x*_Z)-1)^2, x)) [in Maple notation].

%F 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

%p 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

%t 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 *)

%o (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

%Y Cf. A052894.

%K nonn,nice,easy

%O 1,2

%A _Richard Stanley_

%E More terms from _Vladeta Jovovic_, Nov 04 2001