A007453 Number of unlabeled connected series-parallel posets with n nodes. (Formerly M2813)

1, 1, 3, 9, 30, 103, 375, 1400, 5380, 21073, 83950, 338878, 1383576, 5702485, 23696081, 99163323, 417553252, 1767827220, 7520966100, 32135955585, 137849390424, 593407692685, 2562695780058, 11099806544050, 48206136562750, 209876865026303, 915840095739301

Offset

1,3

References

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

Links

Alois P. Heinz, Table of n, a(n) for n = 1..1000 (first 100 terms from Jean-François Alcover)

B. I. Bayoumi, M. H. El-Zahar and S. M. Khamis, Asymptotic enumeration of N-free partial orders, Order 6 (1989), 219-232.

P. J. Cameron, On the probability of connectedness, Discrete Math., 167 (1997), 175-187.

P. J. Cameron, Some sequences of integers, Discrete Math., 75 (1989), 89-102.

P. J. Cameron, Some sequences of integers, in "Graph Theory and Combinatorics 1988", ed. B. Bollobas, Annals of Discrete Math., 43 (1989), 89-102.

Soheir M. Khamis, Height counting of unlabeled interval and N-free posets, Discrete Math. 275 (2004), no. 1-3, 165-175.

Index entries for sequences related to posets

Formula

See the 1989 and 1997 papers by Cameron cited above for generating functions, and the 1997 paper for asymptotics.

Inverse Euler transform of A003430. - Sean A. Irvine, Jan 04 2018

a(n) = A003430(n) - A007454(n) for n > 1. - Sean A. Irvine, Jan 04 2018

Mathematica

terms = 25; A[_] = 1;
Do[A[x_] = Exp[Sum[(1/k)*(A[x^k] + 1/A[x^k] - 2 + x^k), {k, 1, terms+1}]] + O[x]^(terms+1) // Normal, terms+1];
A003430 = CoefficientList[A[x], x] // Rest;
mob[m_, n_] := If[Mod[m, n] == 0, MoebiusMu[m/n], 0];
EULERi[b_] := Module[{a, c, i, d}, c = {}; For[i = 1, i <= Length[b], i++, c = Append[c, i*b[[i]] - Sum[c[[d]]*b[[i - d]], {d, 1, i-1}]]]; a = {}; For[i = 1, i <= Length[b], i++, a = Append[a, (1/i)*Sum[mob[i, d]*c[[d]], {d, 1, i}]]]; Return[a]];
EULERi[A003430] (* Jean-François Alcover, Jan 23 2020 *)

Crossrefs

Cf. A003430, A007454.

Sequence in context: A228248 A047064 A047007 * A148952 A148953 A148954

Adjacent sequences: A007450 A007451 A007452 * A007454 A007455 A007456

Keyword

nonn,nice

Author

N. J. A. Sloane, Mira Bernstein

Extensions

Name corrected by Salah Uddin Mohammad, Jun 07 2020

Status

approved