A005193 a(n) is the number of alpha-labelings of graphs with n edges. (Formerly M1231)

1, 2, 4, 10, 30, 106, 426, 1930, 9690, 53578, 322650, 2106250, 14790810, 111327178, 893091930, 7614236170, 68695024410, 654301474378, 6557096219610, 69005893630090, 760519875693210, 8763511069234378, 105343011537811290, 1319139904954848010

Offset

1,2

Comments

Old name was: Balanced labeled graphs. New name taken from Mar 06 2021 comment from Don Knuth.

References

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

Links

Table of n, a(n) for n=1..24.

C. Barrientos and S. M. Minion, Enumerating families of labeled graphs, J. Integer Seq., 18(2015), #15.1.7.

Henryk Fuks and Kate Sullivan, Enumeration of number-conserving cellular automata rules with two inputs, arXiv:0711.1349 [nlin.CG], 2007; Journal of Cellular Automata 2 vol. 2 pp. 141-148 (2007).

David A. Sheppard, The factorial representation of major balanced labelled graphs, Discrete Math., 15(1976), no. 4, 379-388.

Formula

If n is even then a(n) = 2*Sum_{j=1..floor(n/2)} j!^2*j^(n-2*j), otherwise a(n) = 2*Sum_{j=1..floor(n/2)} j!^2*j^(n-2*j) + ((n+1)/2)!*((n-1)/2)!. - Jonathan Vos Post, Nov 13 2007

Maple

A005193 := proc(q)
    2*add((j!)^2*j^(q-2*j), j=1..q/2) ;
    if type(q, 'odd') then
        %+((q+1)/2)!*((q-1)/2)! ;
    else
        % ;
    end if;
end proc:
seq(A005193(n), n=1..40) ; # R. J. Mathar, Jul 13 2014

Mathematica

a[n_] := 2 Sum[(j!)^2*j^(n-2j), {j, 1, n/2}] + Boole[OddQ[n]]*((n+1)/2)! * ((n-1)/2)!;
Array[a, 24] (* Jean-François Alcover, Nov 20 2017 *)

Crossrefs

Cf. A034384, A342225.

Sequence in context: A186021 A332650 A091174 * A317280 A173940 A101901

Adjacent sequences: A005190 A005191 A005192 * A005194 A005195 A005196

Keyword

nonn

Author

N. J. A. Sloane

Extensions

Renamed (using Comments entry from Don Knuth) by Jon E. Schoenfield, Oct 28 2023

Status

approved