%I M3061 #74 Jun 17 2022 16:17:07

%S 1,1,3,19,183,2371,38703,763099,17648823,468603091,14050842303,

%T 469643495179,17315795469063,698171064855811,30561156525545103,

%U 1443380517590979259,73161586346500098903,3961555049961803092531,228225249142441259147103,13938493569348563803135339

%N Semiorders on n elements.

%C Labeled semiorders on n elements: (1+3) and (2+2)-free posets. - Detlef Pauly (dettodet(AT)yahoo.de), Dec 27 2002

%C Labeled incomplete binary trees (every vertex has a left child, a right child, neither, or both) in which every vertex with a right child but no left child has a label greater than the label of its right child. - _Ira M. Gessel_, Nov 09 2013

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%D R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 6.30.

%H Vincenzo Librandi, <a href="/A006531/b006531.txt">Table of n, a(n) for n = 0..200</a>

%H J. L. Chandon, J. LeMaire and J. Pouget, <a href="http://www.numdam.org/item?id=MSH_1978__62__61_0">Dénombrement des quasi-ordres sur un ensemble fini</a>, Math. Sci. Humaines, No. 62 (1978), 61-80.

%H J. L. Chandon, J. LeMaire and J. Pouget, <a href="/A006531/a006531_1.pdf">Enumeration of semiorders on a finite set</a>, Preprint (English) of "Dénombrement des quasi-ordres sur un ensemble fini".

%H J. L. Chandon, <a href="/A006531/a006531.pdf">Letter to N. J. A. Sloane, May 1981</a>

%H Julie Christophe, Jean-Paul Doignon and Samuel Fiorini, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL6/Doignon/doignon40.html">Counting Biorders</a>, J. Integer Seqs., Vol. 6, 2003.

%H Yan X Zhang, <a href="http://arxiv.org/abs/1508.00318">Four Variations on Graded Posets</a>, arXiv preprint arXiv:1508.00318 [math.CO], 2015.

%F E.g.f.: C(1-exp(-x)), where C(x) = (1 - sqrt(1 - 4*x)) / (2*x) is the ordinary g.f. for the Catalan numbers A000108. [corrected by _Joel B. Lewis_, Mar 29 2011]

%F a(n) = Sum_{k=1..n} S(n, k) * k! * M(k-1), S(n, k): Stirling number of the second kind, M(n): Motzkin number, A001006. - Detlef Pauly, Jun 06 2002

%F O.g.f.: Sum_{n>=1} (2*n)!/(n+1)! * x^n / Product_{k=0..n} (1+k*x). - _Paul D. Hanna_, Jul 20 2011

%F a(n) ~ n! * sqrt(3)*(log(4/3))^(1/2-n)/(sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Aug 13 2013

%F E.g.f.: 1/(1 + (exp(-x) - 1)/(1 + (exp(-x) - 1)/(1 + (exp(-x) - 1)/(1 + (exp(-x) - 1)/(1 + ...))))), a continued fraction. - _Ilya Gutkovskiy_, Nov 18 2017

%F From _Peter Bala_, Jan 15 2018: (Start)

%F a(n) = Sum_{k = 0..n} (-1)^(n+k)*Catalan(k)*k!*Stirling2(n,k). Cf. A052895.

%F Conjecture: for fixed k = 1,2,..., the sequence a(n) (mod k) is eventually periodic with the exact period dividing phi(k), where phi(k) is Euler's totient function A000010. For example, modulo 10 the sequence becomes (1, 1, 3, 9, 3, 1, 3, 9, 3, ...) with an apparent period 1, 3, 9, 3 of length 4 = phi(10) beginning at a(1). (End)

%F Consider the transformation of a sequence u given by T(u)(m) = (-1)^m*Sum_{n=0..m} (u(n)/(n+1))*(Sum_{k=0..n}(-1)^k*binomial(n,k)*k^m). If u(n) = 1 then T(u)(n) = Bernoulli(n) (with Bernoulli(1) = 1/2), if u(n) = binomial(2*n,n) then T(u)(n) = a(n). - _Peter Luschny_, Jul 09 2020

%p A006531 := n->add(stirling2(n,k)*k!*A001006(k-1),k=1..n);

%t m[n_] := m[n] = m[n-1] + Sum[ m[k]*m[n-k-2], {k, 0, n-2}]; m[0] = a[0] = 1; a[n_] := Sum[ StirlingS2[n, k]*k!*m[k-1], {k, 1, n}]; Table[a[n], {n, 0, 17}] (* _Jean-François Alcover_, Jul 24 2012, after Maple *)

%o (PARI) {a(n)=polcoeff(sum(m=0, n, (2*m)!/(m+1)!*x^m/prod(k=1, m, 1+k*x+x*O(x^n))), n)} /* _Paul D. Hanna_, Jul 20 2011 */

%Y Cf. A000108 (unlabeled semiorders: Catalan numbers), A052895.

%K nonn,nice,easy

%O 0,3

%A _N. J. A. Sloane_