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Number of semiorders on n labeled nodes whose incomparability graph is connected.
11

%I #34 Jun 13 2021 16:45:02

%S 1,1,7,61,751,11821,226927,5142061,134341711,3975839341,131463171247,

%T 4803293266861,192178106208271,8356430510670061,392386967808249967,

%U 19788154572706556461,1066668756919315412431,61204224384073232815981

%N Number of semiorders on n labeled nodes whose incomparability graph is connected.

%H Robert Israel, <a href="/A048287/b048287.txt">Table of n, a(n) for n = 1..373</a>

%H Mats Granvik, <a href="https://math.stackexchange.com/questions/585942/">Power series to calculate Lambert W up to infinity</a>

%F E.g.f.: 1-2*(1-exp(-x))/(1-sqrt(4*exp(-x)-3)).

%F E.g.f.: (1 - sqrt(4*exp(-x) - 3)) / 2. - _Michael Somos_, Nov 26 2017

%F a(n) = Sum_{k=1..n} (-1)^(n-k)*Stirling2(n, k)*k!*Catalan(k-1). - _Vladeta Jovovic_, Oct 18 2003

%F Equals column 1 (unsigned) of triangle A136595, which is the matrix inverse of the triangle A136590 of trinomial logarithmic coefficients. - _Paul D. Hanna_, Jan 10 2008

%F E.g.f A(x)=F(exp(x)-1), F(x)=x*A005043(x). - _Vladimir Kruchinin_, Sep 07 2010

%F a(n) = (-1)^(n-1) + Sum_{k=1..n-1} binomial(n,k)*a(k)*a(n-k). - _Robert Israel_, Mar 01 2016

%F Given e.g.f. =: A(x), then exp(-x) = A(x)^2 - A(x) + 1 = A'(x)*(1 - 2*A(x)). - _Michael Somos_, Nov 26 2017

%F a(n) ~ sqrt(3/8) * n^(n-1) / (log(4/3)^(n - 1/2) * exp(n)). - _Vaclav Kotesovec_, Dec 16 2020

%e a(3)=7, the seven semiorders being three disjoint points and the disjoint union of a point and a two-element chain (with six labelings).

%p A048287 := n -> add((-1)^(n-k-1)*Stirling2(n,k+1)*(2*k)!/k!, k=0..n-1):

%p seq(A048287(n), n=1..18); # _Peter Luschny_, Jan 27 2016

%t Table[Sum[(-1)^(n - k) StirlingS2[n, k] k!*CatalanNumber[k - 1], {k, n}], {n, 20}] (* _Michael De Vlieger_, Jan 27 2016 *)

%t Rest[Range[0, 18]! CoefficientList[Series[1 - 2 (1 - Exp[-x]) /(1 - Sqrt[4 Exp[-x] - 3]), {x, 0, 18}], x]] (* _Vincenzo Librandi_, Jan 28 2016 *)

%o (PARI) {a(n)=local(A136590=matrix(n+1,n+1,r,c,if(r>=c,(r-1)!/(c-1)!*polcoeff(log(1+x+x^2 +x*O(x^n))^(c-1),r-1))));(-1)^(n+1)*(A136590^-1)[n+1,2]} \\ _Paul D. Hanna_, Jan 10 2008

%o (PARI) {a(n) = if( n<0, 0, n! * polcoeff( (1 - sqrt(4*exp(-x + x*O(x^n)) - 3)) / 2, n))}; /* _Michael Somos_, Nov 26 2017 */

%o (PARI) {a(n) = if( n<1, 0, n! * polcoeff( serreverse( -log(1 - x + x^2 + x * O(x^n))), n))}; /* _Michael Somos_, Nov 26 2017 */

%Y Cf. A000108, A006531.

%Y Cf. A136595, A136590.

%K easy,nonn

%O 1,3

%A _Richard Stanley_

%E More terms from _Vladeta Jovovic_, Oct 18 2003