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A048287 Number of semiorders on n labeled nodes whose incomparability graph is connected. 10
1, 1, 7, 61, 751, 11821, 226927, 5142061, 134341711, 3975839341, 131463171247, 4803293266861, 192178106208271, 8356430510670061, 392386967808249967, 19788154572706556461, 1066668756919315412431, 61204224384073232815981 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

LINKS

Robert Israel, Table of n, a(n) for n = 1..373

FORMULA

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

a(n) = Sum_{k=1..n} (-1)^(n-k)*Stirling2(n, k)*k!*Catalan(k-1). - Vladeta Jovovic, Oct 18 2003

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

E.g.f A(x)=F(exp(x)-1), F(x)=x*A005043(x). - Vladimir Kruchinin, Sep 07 2010

a(n) = (-1)^(n-1) + Sum_{1<=k<=n-1} binomial(n,j)*a(j)*a(n-j). - Robert Israel, Mar 01 2016

EXAMPLE

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).

MAPLE

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

seq(A048287(n), n=1..18); # Peter Luschny, Jan 27 2016

MATHEMATICA

Table[Sum[(-1)^(n - k) StirlingS2[n, k] k!*CatalanNumber[k - 1], {k, n}], {n, 20}] (* Michael De Vlieger, Jan 27 2016 *)

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 *)

PROG

(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

CROSSREFS

Cf. A000108, A006531.

Cf. A136595, A136590.

Sequence in context: A001830 A213326 A261901 * A145507 A254121 A047685

Adjacent sequences:  A048284 A048285 A048286 * A048288 A048289 A048290

KEYWORD

easy,nonn

AUTHOR

Richard Stanley

EXTENSIONS

More terms from Vladeta Jovovic, Oct 18 2003

STATUS

approved

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Last modified May 4 20:53 EDT 2016. Contains 272418 sequences.