A056290
Number of primitive (period n) n-bead necklaces with exactly five different colored beads.
Original entry on oeis.org
0, 0, 0, 0, 24, 300, 2400, 15750, 92680, 510288, 2691600, 13793850, 69309240, 343499100, 1686135352, 8221421250, 39901776360, 193053923860, 932142850800, 4495236287850, 21664357532920, 104388118174500, 503044634004000, 2425003910574000, 11696087875731600
Offset: 1
- M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]
-
with(numtheory):
b:= proc(n, k) option remember; `if`(n=0, 1,
add(mobius(n/d)*k^d, d=divisors(n))/n)
end:
a:= n-> add(b(n, 5-j)*binomial(5, j)*(-1)^j, j=0..5):
seq(a(n), n=1..30); # Alois P. Heinz, Jan 25 2015
-
b[n_, k_] := b[n, k] = If[n==0, 1, DivisorSum[n, MoebiusMu[n/#]*k^# &]/n];
a[n_] := Sum[b[n, 5 - j]*Binomial[5, j]*(-1)^j, {j, 0, 5}];
Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Jun 06 2018, after Alois P. Heinz *)
A056291
Number of primitive (period n) n-bead necklaces with exactly six different colored beads.
Original entry on oeis.org
0, 0, 0, 0, 0, 120, 2160, 23940, 211680, 1643544, 11748240, 79419060, 516257280, 3262440960, 20193277104, 123071683140, 741419995680, 4427489935680, 26264144909520, 155018839412052, 911509010152560, 5344538372696880, 31272099902089200, 182707081042818360
Offset: 1
- M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]
-
with(numtheory):
b:= proc(n, k) option remember; `if`(n=0, 1,
add(mobius(n/d)*k^d, d=divisors(n))/n)
end:
a:= n-> add(b(n, 6-j)*binomial(6, j)*(-1)^j, j=0..6):
seq(a(n), n=1..30); # Alois P. Heinz, Jan 25 2015
-
b[n_, k_] := b[n, k] = If[n==0, 1, DivisorSum[n, MoebiusMu[n/#]*k^# &]/n];
a[n_] := Sum[b[n, 6 - j]*Binomial[6, j]*(-1)^j, {j, 0, 6}];
Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Jun 06 2018, after Alois P. Heinz *)
A317852
Number of plane trees with n nodes where the sequence of branches directly under any given node is aperiodic, meaning its cyclic permutations are all different.
Original entry on oeis.org
1, 1, 1, 3, 8, 26, 76, 247, 783, 2565, 8447, 28256, 95168, 323720, 1108415, 3821144, 13246307, 46158480, 161574043, 567925140, 2003653016, 7092953340, 25186731980, 89690452750, 320221033370, 1146028762599, 4110596336036, 14774346783745, 53203889807764, 191934931634880
Offset: 1
The a(5) = 8 locally aperiodic plane trees:
((((o)))),
(((o)o)), ((o(o))), (((o))o), (o((o))),
((o)oo), (o(o)o), (oo(o)).
The a(6) = 26 locally aperiodic plane trees:
(((((o))))) ((((o)o))) (((o)oo)) ((o)ooo)
(((o(o)))) ((o(o)o)) (o(o)oo)
((((o))o)) ((oo(o))) (oo(o)o)
((o((o)))) (((o)o)o) (ooo(o))
((((o)))o) ((o(o))o)
(o(((o)))) (o((o)o))
(((o))(o)) (o(o(o)))
((o)((o))) (((o))oo)
(o((o))o)
(oo((o)))
((o)(o)o)
((o)o(o))
(o(o)(o))
Cf.
A000108,
A000837,
A007853,
A032171,
A032200,
A254040,
A301700,
A303386,
A303431,
A304173,
A304175,
A317708,
A317852.
-
aperQ[q_]:=Array[RotateRight[q,#]&,Length[q],1,UnsameQ];
aperplane[n_]:=If[n==1,{{}},Join@@Table[Select[Tuples[aperplane/@c],aperQ],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[aperplane[n]],{n,10}]
-
Tfm(p, n)={sum(d=1, n, moebius(d)*(subst(1/(1+O(x*x^(n\d))-p), x, x^d)-1))}
seq(n)={my(p=O(1)); for(i=1, n, p=1+Tfm(x*p, i)); Vec(p)} \\ Andrew Howroyd, Feb 08 2020
A254079
Number of primitive (=aperiodic) n-bead necklaces with colored beads of exactly 7 different colors.
Original entry on oeis.org
720, 17640, 258720, 2963520, 29317680, 263506320, 2215825920, 17758540440, 137337087888, 1033793011440, 7621696948320, 55289495222640, 396017903136240, 2808175618022544, 19754629731587280, 138087381577881960, 960370246763096400, 6652267408747194360
Offset: 7
-
with(numtheory):
b:= proc(n, k) option remember; `if`(n=0, 1,
add(mobius(n/d)*k^d, d=divisors(n))/n)
end:
a:= n-> add(b(n, 7-j)*binomial(7, j)*(-1)^j, j=0..7):
seq(a(n), n=7..30);
A254080
Number of primitive (=aperiodic) n-bead necklaces with colored beads of exactly 8 different colors.
Original entry on oeis.org
5040, 161280, 3024000, 43545600, 534330720, 5891719680, 60227481600, 582295633920, 5397245411040, 48421936442880, 423440785541760, 3628271603174400, 30584600246448864, 254421149466401280, 2093705950217414400, 17078070713147136000, 138294714025711281360
Offset: 8
-
with(numtheory):
b:= proc(n, k) option remember; `if`(n=0, 1,
add(mobius(n/d)*k^d, d=divisors(n))/n)
end:
a:= n-> add(b(n, 8-j)*binomial(8, j)*(-1)^j, j=0..8):
seq(a(n), n=8..30);
A254081
Number of primitive (=aperiodic) n-bead necklaces with colored beads of exactly 9 different colors.
Original entry on oeis.org
40320, 1632960, 38102400, 673596000, 10035083520, 133102569600, 1623972430080, 18615386790000, 203401119841920, 2140495978400640, 21860934514473600, 217932712305073920, 2130148393318725120, 20485620162998625600, 194378546540211264000, 1823813323809879402000
Offset: 9
-
with(numtheory):
b:= proc(n, k) option remember; `if`(n=0, 1,
add(mobius(n/d)*k^d, d=divisors(n))/n)
end:
a:= n-> add(b(n, 9-j)*binomial(9, j)*(-1)^j, j=0..9):
seq(a(n), n=9..30);
A254082
Number of primitive (=aperiodic) n-bead necklaces with colored beads of exactly 10 different colors.
Original entry on oeis.org
362880, 18144000, 515592000, 10977120000, 195113318400, 3063348288000, 43943631732000, 588790762560000, 7481812222684800, 91158709273632000, 1073686615986821760, 12301136459932320000, 137753173599205449600, 1513588462073525376000, 16368017165881385004000
Offset: 10
-
with(numtheory):
b:= proc(n, k) option remember; `if`(n=0, 1,
add(mobius(n/d)*k^d, d=divisors(n))/n)
end:
a:= n-> add(b(n, 10-j)*binomial(10, j)*(-1)^j, j=0..10):
seq(a(n), n=10..30);
A254083
Number of primitive (=aperiodic) 2n-bead necklaces with colored beads of exactly n different colors.
Original entry on oeis.org
1, 0, 3, 89, 5100, 510288, 79419060, 17758540440, 5397245411040, 2140495978400640, 1073686615986821760, 664582969579045104000, 497566995304189636425600, 443212653988584641970547200, 463237380681508395317004249600, 561422444732790213860667834854400
Offset: 0
a(2) = 3: 0001, 0011, 0111.
-
with(numtheory):
b:= proc(n, k) option remember; `if`(n=0, 1,
add(mobius(n/d)*k^d, d=divisors(n))/n)
end:
a:= n-> add(b(2*n, n-j)*binomial(n, j)*(-1)^j, j=0..n):
seq(a(n), n=0..20);
-
b[n_, k_] := b[n, k] = If[n==0, 1, DivisorSum[n, MoebiusMu[n/#]*k^#&]/n]; a[n_] := Sum[b[2n, n-j]*Binomial[n, j]*(-1)^j, {j, 0, n}]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Feb 26 2017, translated from Maple *)
Comments