A151879 Produced by same formula that gives A000568 (unlabeled tournaments), but with LCM instead of GCD in the exponent.

1, 1, 1, 2, 8, 52, 528, 8632, 252928, 15494032, 2050181376, 525675623520, 239430803636224, 189133678584246592, 260786292437892272128, 638374284463941710477184, 2842966981002836533300953088, 23866119110542723640161098330368, 394851495657676102988098496313229312

Offset: 0

Author

N. J. A. Sloane, Jul 21 2009

Keywords

• nonn

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..50

Programs

Mathematica

permcount[v_] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t*k; s += t]; s!/m];
edges[v_] := Sum[Sum[LCM[v[[i]], v[[j]]], {j, 1, i - 1}], {i, 2, Length[v]} ] + Sum[Quotient[v[[i]], 2], {i, 1, Length[v]}];
oddp[v_] := (For[i = 1, i <= Length[v], i++, If[BitAnd[v[[i]], 1] == 0, Return[0]]]; 1);
a[n_] := a[n] = (s = 0; Do[If[oddp[p] == 1, s += permcount[p]*2^edges[p]], {p, IntegerPartitions[n]}]; s/n!); (* Jean-François Alcover, Nov 13 2017, after Andrew Howroyd *)

PARI

permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
edges(v) = {sum(i=2, #v, sum(j=1, i-1, lcm(v[i], v[j]))) + sum(i=1, #v, v[i]\2)}
oddp(v) = {for(i=1, #v, if(bitand(v[i], 1)==0, return(0))); 1}
a(n) = {my(s=0); forpart(p=n, if(oddp(p), s+=permcount(p)*2^(edges(p)))); s/n!} \\ Andrew Howroyd, Feb 29 2020

Python

from math import prod, lcm, factorial
from fractions import Fraction
from itertools import product
from sympy.utilities.iterables import partitions
def A151879(n): return int(sum(Fraction(1<<(sum(p[r]*p[s]*lcm(r,s) for r,s in product(p.keys(),repeat=2))-sum(p.values())>>1),prod(q**p[q]*factorial(p[q]) for q in p)) for p in partitions(n) if all(q&1 for q in p))) # Chai Wah Wu, Jul 01 2024

Formula

a(n) = Sum_{j} (1/(Product (k^(j_k) (j_k)!))) * 2^{t_j}, where j runs through all partitions of n into odd parts, say with j_1 parts of size 1, j_3 parts of size 3, etc., and t_j = (1/2)*[ Sum_{r=1..n, s=1..n} j_r j_s lcm(r,s) - Sum_{r} j_r ].