A061686 -id:A061686 - cp's OEIS frontend

A091112 Number of orbits of length n under the map whose periodic points are counted by A061686.

1, 8, 513, 115272, 70162625, 95640604266, 256797561193432, 1238094271228829120, 9993778343964199218438, 127849400250667505250954500, 2480163309080566931933236667234, 70354340598798824605743590305386600, 2830805474672999382519296750329811657242

Offset: 1

Thomas Ward, Feb 24 2004

nonn

Old Name was: "A061686 appears to count the periodic points for a certain map. If so, then this is the sequence of the numbers of orbits of length n under that map".

			b(1)=1, b(3)=1540, so a(3)=(1/3)(b(3)-b(1))=513.

Robert Israel, Table of n, a(n) for n = 1..126
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].

Cf. A061686.

a061686:= proc(n) option remember;
  add(binomial(n,k)^5*(n-k)*procname(k)/n, k=0..n-1)
end proc:
a061686(0):= 1:
a:= n -> 1/n * add(numtheory:-mobius(d)*a061686(n/d), d = numtheory:-divisors(n)):
seq(a(n), n=1..6); # Robert Israel, May 05 2015

(* b = A061686 *) b[0]=1; b[n_] := b[n] = Sum[Binomial[n, k]^5*(n-k)*b[k]/ n, {k, 0, n-1}]; a[n_] := (1/n)*DivisorSum[n, MoebiusMu[#] * b[n/#] &]; Array[a, 20] (* Jean-François Alcover, Dec 04 2015 *)

A091112(n)=sumdiv(n,d,moebius(d)*A061686(n/d)) \\ M. F. Hasler, May 11 2015

If b(n) is the (n+1)-th term of A061686, then a(n) = (1/n)*Sum_{d|n} mu(d) b(n/d).

More terms from Robert Israel, May 05 2015

Name clarified by M. F. Hasler, May 11 2015

A275043 Number A(n,k) of set partitions of [k*n] such that within each block the numbers of elements from all residue classes modulo k are equal for k>0, A(n,0)=1; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 5, 1, 1, 1, 5, 16, 15, 1, 1, 1, 9, 64, 131, 52, 1, 1, 1, 17, 298, 1613, 1496, 203, 1, 1, 1, 33, 1540, 25097, 69026, 22482, 877, 1, 1, 1, 65, 8506, 461105, 4383626, 4566992, 426833, 4140, 1, 1, 1, 129, 48844, 9483041, 350813126, 1394519922, 437665649, 9934563, 21147, 1

Offset: 0

Alois P. Heinz, Jul 14 2016

			A(2,2) = 3: 1234, 12|34, 14|23.
A(2,3) = 5: 123456, 123|456, 126|345, 135|246, 156|234.
A(2,4) = 9: 12345678, 1234|5678, 1238|4567, 1247|3568, 1278|3456, 1346|2578, 1368|2457, 1467|2358, 1678|2345.
A(3,2) = 16: 123456, 1234|56, 1236|45, 1245|36, 1256|34, 12|3456, 12|34|56, 12|36|45, 1346|25, 1456|23, 14|2356, 14|23|56, 16|2345, 16|23|45, 14|25|36, 16|25|34.
Square array A(n,k) begins:
  1,   1,     1,       1,          1,            1,               1, ...
  1,   1,     1,       1,          1,            1,               1, ...
  1,   2,     3,       5,          9,           17,              33, ...
  1,   5,    16,      64,        298,         1540,            8506, ...
  1,  15,   131,    1613,      25097,       461105,         9483041, ...
  1,  52,  1496,   69026,    4383626,    350813126,     33056715626, ...
  1, 203, 22482, 4566992, 1394519922, 573843627152, 293327384637282, ...

Alois P. Heinz, Antidiagonals n = 0..60, flattened
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.
Wikipedia, Partition of a set

Columns k=0-10 give: A000012, A000110, A023998, A061684, A061685, A061686, A061687, A061688, A275097, A275098, A275099.
Rows n=0+1,2-5 give: A000012, A094373, A275100, A275101, A275102.
Main diagonal gives A275044.
Cf. A345400.

A:= proc(n, k) option remember; `if`(k*n=0, 1, add(
       binomial(n, j)^k*(n-j)*A(j, k), j=0..n-1)/n)
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);

A[n_, k_] := A[n, k] = If[k*n == 0, 1, Sum[Binomial[n, j]^k*(n-j)*A[j, k], {j, 0, n-1}]/n]; Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Jan 17 2017, translated from Maple *)

A342199 a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k)^5 * a(k-1).

Original entry on oeis.org

1, 1, 33, 519, 43111, 5068111, 840782023, 291086377719, 139698959369111, 90748115988081551, 90809507057803456103, 124011515918275951611959, 217278911997171247450862041, 509237348184229328050319432621, 1567286639251140454692258569881053

Offset: 0

Ilya Gutkovskiy, Mar 04 2021

nonn

Cf. A040027, A061686, A336197, A342196, A342197, A342198.

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k]^5 a[k - 1], {k, 1, n}]; Table[a[n], {n, 0, 14}]

cp's OEIS Frontend

A091112 Number of orbits of length n under the map whose periodic points are counted by A061686.

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A275043 Number A(n,k) of set partitions of [k*n] such that within each block the numbers of elements from all residue classes modulo k are equal for k>0, A(n,0)=1; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A342199 a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k)^5 * a(k-1).

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