A000662 Number of relations with 3 arguments on n nodes. (Formerly M2180 N0872)

2, 136, 22377984, 768614354122719232, 354460798875983863749270670915141632, 146267071761884981524915186989628577728537526896649216991428608

Offset

1,1

References

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, p. 76 (2.2.31)

W. Oberschelp, Kombinatorische Anzahlbestimmungen in Relationen, Math. Ann., 174 (1967), 53-78.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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..15

P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.

W. Oberschelp, Strukturzahlen in endlichen Relationssystemen, in Contributions to Mathematical Logic (Proceedings 1966 Hanover Colloquium), pp. 199-213, North-Holland Publ., Amsterdam, 1968. [Annotated scanned copy]

Formula

a(n) = Sum_{1*s_1+2*s_2+...=n} (fixA[s_1, s_2,...]/(1^s_1*s_1!*2^s_2*s_2!*...)) where fixA[s_1, s_2, ...] = 2^Sum_{i, j, k>=1} (i*j*k*s_i*s_j*s_k/lcm(i, j, k)). - Christian G. Bower, Jan 06 2004

Prog

(Python)
from itertools import product
from math import factorial, prod, lcm
from fractions import Fraction
from sympy.utilities.iterables import partitions
def A000662(n): return int(sum(Fraction(1<<sum(r*s*t//lcm(r, s, t)*p[r]*p[s]*p[t] for r, s, t in product(p.keys(), repeat=3)), prod(q**p[q]*factorial(p[q]) for q in p)) for p in partitions(n))) # Chai Wah Wu, Jul 02 2024

Crossrefs

Cf. A000595, A001377, A051241.

Sequence in context: A226118 A216030 A216083 * A139907 A195379 A221227

Adjacent sequences: A000659 A000660 A000661 * A000663 A000664 A000665

Keyword

nonn,nice

Author

N. J. A. Sloane

Status

approved