A370459
Number of unicursal stars with n vertices.
Original entry on oeis.org
0, 0, 1, 1, 5, 19, 112, 828, 7441, 76579, 871225, 10809051, 144730446, 2079635889, 31912025537, 520913578812, 9013780062785, 164829273635749, 3176388519597555, 64343477504391475, 1366925655386979893, 30390554390984325019, 705740995420852895453
Offset: 3
For n=5, there is only the regular pentagram {5/2}.
For n=6, there is only the unicursal hexagram.
For n=7, in addition to the two regular heptagrams {7/2} and {7/3}, there are three nontrivial unicursal heptagrams represented by:
(0, 2, 4, 1, 6, 3, 5, 0)
(0, 2, 5, 1, 3, 6, 4, 0)
(0, 2, 5, 1, 4, 6, 3, 0).
Cf.
A000940 (polygon sides allowed).
Cf.
A055684 (cases with dihedral symmetry only).
Cf.
A002816 (rotations and reflections counted separately).
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\\ Requires a370068 from A370068.
Ro(n)=-(-1)^n + subst(serlaplace(polcoef(((1 - x)^2)/(2*(1 + x)*(1 + (1 - 2*y)*x + 2*y*x^2)) + O(x*x^n), n)), y, 1)
Re(n)=subst(serlaplace(polcoef((1 - x - 2*x^2)/(4*(1 + (1 - 2*y)*x + 2*y*x^2)) + O(x*x^n), n)), y, 1)
a(n)={if(n<3, 0, (if(n%2, 2*Ro(n\2), Re(n/2)) + a370068(n))/4)} \\ Andrew Howroyd, Mar 01 2024
A003224
The number of superpositions of cycles of order n of the groups E_3 and D_n.
Original entry on oeis.org
1, 5, 24, 391, 9549, 401547, 22597671, 1646431048, 149640359575, 16597459048676, 2206178465445432, 346212403086248325, 63333787189956042080, 13359470726804093346852, 3218846593376516669825536, 878566295178157438213870011
Offset: 3
- F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 171.
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
A384631
Number of self-inverse double cosets in D_n\S_n/D_n.
Original entry on oeis.org
1, 2, 4, 8, 17, 52, 153, 482, 1623, 5879, 21926, 85436, 344998, 1444437, 6230232, 27704051, 126571091, 593974930, 2856031804, 14065575098, 70822693101, 364420818168, 1913609207886, 10249715874962, 55938458263035, 310915671908063, 1758452185453926, 10115287840489764
Offset: 3
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# From Proposition 4.2 in the reference:
from sympy import divisors, factorial, totient
def A384631(n):
s = 0
if n%2==0:
for d in divisors(n//2):
if d%2==0:
s += totient(d)*factorial(n//d)*(d//2)**(n//2//d)//factorial(n//2//d)
else:
s += totient(d)*sum(factorial(n//d)*d**i//2**i//factorial(i)//factorial(n//d-2*i) for i in range(n//2//d+1))
s += n//2*((n%4)//2+1)*factorial(2*(n//4))//factorial(n//4)
else:
for d in divisors(n):
s += totient(d)*sum(factorial(n//d)*d**i//2**i//factorial(i)//factorial(n//d-2*i) for i in range(n//d//2+1))
if n%4==1:
s += n*factorial(n//2)//factorial(n//4)
return s//n//2
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