A091160
Number of orbits of length n under the map whose periodic points are counted by A061687.
Original entry on oeis.org
1, 16, 2835, 2370752, 6611343125, 48887897438124, 821067869874486556, 28006755051982013513984, 1782755223314276717178818904, 198173677662343700104263938337400, 36467946245662764068249155883368682252, 10631160782054640951386529213624176084501136
Offset: 1
b(1)=1, b(3)=8506, so a(3) = (1/3)*(8506-1) = 2835.
- Alois P. Heinz, Table of n, a(n) for n = 1..100
- Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
- J.-M. Sixdeniers, K. A. Penson and A. I. Solomon, Extended Bell and Stirling Numbers From Hypergeometric Exponentiation, J. Integer Seqs. Vol. 4 (2001), #01.1.4.
- Thomas Ward, Exactly realizable sequences. [local copy].
-
with(numtheory):
b:= proc(n) option remember;
`if`(n=0, 1, add(binomial(n, k)^6*(n-k)*b(k)/n, k=0..n-1))
end:
a:= n-> add(mobius(d)*b(n/d), d=divisors(n))/n:
seq(a(n), n=1..15); # Alois P. Heinz, Mar 19 2014
-
b[n_] := b[n] = If[n==0, 1, Sum[Binomial[n, k]^6 (n-k)b[k]/n, {k, 0, n-1}]];
a[n_] := Sum[MoebiusMu[d] b[n/d], {d, Divisors[n]}]/n;
Array[a, 15] (* Jean-François Alcover, Nov 18 2020, after Alois P. Heinz *)
A091266
Number of orbits of length n under the map whose periodic points are counted by A061694.
Original entry on oeis.org
0, 0, 12, 216, 3500, 58494, 1028167, 18954072, 363991752, 7231521650, 147777013109, 3091874792274, 65993049570175, 1432803420182428, 31570847522072400, 704668366087255200, 15907964778448807820
Offset: 1
b(1)=0, b(3)=36 so a(3)=12.
- Vaclav Kotesovec, Table of n, a(n) for n = 1..200
- Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
- J.-M. Sixdeniers, K. A. Penson and A. I. Solomon, Extended Bell and Stirling Numbers From Hypergeometric Exponentiation, J. Integer Seqs. Vol. 4 (2001), #01.1.4.
- Thomas Ward, Exactly realizable sequences. [local copy].
-
Table[Sum[MoebiusMu[d] * Sum[Sum[((n/d)!/(i!*j!*(n/d - i - j)!))^3/6, {i, 1, n/d - j - 1}], {j, 1, n/d}], {d, Divisors[n]}]/n, {n, 1, 20}] (* Vaclav Kotesovec, Sep 05 2019 *)
A091201
Number of orbits of length n under the map whose periodic points are counted by A061688.
Original entry on oeis.org
1, 32, 16281, 52293792, 692825815625, 28927809504181734
Offset: 1
b(1)=1,b(3)=48844, so a(3)=(1/3)(48844-1)=16281.
A091268
Number of orbits of length n under the map whose periodic points are counted by A061685.
Original entry on oeis.org
1, 4, 99, 6272, 876725, 232419936, 105471170140, 76095730062464, 82555139387847312, 128928209221144677400, 279860608037771819829980, 820360089598849358326307904, 3169977309466844379463315722484
Offset: 1
b(1)=1, b(2)=9, b(3)=298. Hence a(3)=(1/3)(b(3)-b(1))=99.
A091315
Number of orbits of length n under the map whose periodic points are counted by A061684.
Original entry on oeis.org
1, 2, 21, 402, 13805, 761154, 62523664, 7237970648, 1132600004910, 231900134422880, 60528794385067778, 19713593779259862624, 7869483395065035685162, 3792402572391137423764584
Offset: 1
The sequence A061684 begins 1,1,5,64,1613, so a(3)=(b(3)-b(1))/3=21.
A092239
Number of orbits of length n under the map whose periodic points are counted by A061693.
Original entry on oeis.org
0, 2, 9, 42, 225, 1260, 7497, 46176, 293382, 1908150, 12655269, 85287870, 582628683, 4026368514, 28104231825, 197884340160, 1404038987577, 10029929788566, 72086075552493, 520920674929650
Offset: 1
a(3)=9 since a(3)=(1/3)(b(3)-b(1)) where b is the sequence A061693, which starts 0,4,27.
-
Table[Sum[MoebiusMu[d] * (Sum[Binomial[n/d, k]^3, {k, 0, n/d}]/2 - 1), {d, Divisors[n]}]/n, {n, 1, 20}] (* Vaclav Kotesovec, Sep 05 2019 *)
Showing 1-6 of 6 results.
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