A135312 Number of transitive reflexive binary relations R on n labeled elements where |{y : xRy}| <= 2 for all x.

1, 1, 4, 13, 62, 311, 1822, 11593, 80964, 608833, 4910786, 42159239, 383478988, 3678859159, 37087880754, 391641822541, 4319860660448, 49647399946049, 593217470459314, 7354718987639959, 94445777492433516, 1254196823154143191, 17198114810490326714, 243191242578584519333

Offset: 0

Author

Alois P. Heinz, Dec 05 2007

Keywords

• nonn

Examples

			a(2) = 4 because there are 4 relations of the given kind for 2 elements: 1R1, 2R2;  1R1, 2R2, 1R2;  1R1, 2R2, 2R1;  1R1, 2R2, 1R2, 2R1.
		

References

• A. P. Heinz (1990). Analyse der Grenzen und Möglichkeiten schneller Tableauoptimierung. PhD Thesis, Albert-Ludwigs-Universität Freiburg, Freiburg i. Br., Germany.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200

Crossrefs

Column k=2 of A135302.

Cf. A006882, A000248, A007318.

Programs

Maple

u:= proc(n) option remember; add(binomial(n, i)*(n-i)^i, i=0..n) end:
a:= n-> add(binomial(n, 2*i)*doublefactorial(2*i-1)*u(n-2*i), i=0..iquo(n, 2)):
seq(a(n), n=0..50);

Mathematica

a[n_] := SeriesCoefficient[Exp[x*Exp[x] + x^2/2], {x, 0, n}]*n!; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Feb 04 2014 *)

Formula

a(n) = Sum_{i=0..floor(n/2)} C(n,2*i) * A006882(2*i-1) * A000248(n-2*i).

a(n) = A135302(n,2).

E.g.f.: exp(x*exp(x) + x^2/2).