A061688 -id:A061688 - cp's OEIS frontend

A091201 Number of orbits of length n under the map whose periodic points are counted by A061688.

1, 32, 16281, 52293792, 692825815625, 28927809504181734

Offset: 1

Thomas Ward, Feb 24 2004

nonn

Old name was: A061688 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)=48844, so a(3)=(1/3)(48844-1)=16281.

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. A061688.

If b(n) is the (n+1)th term of A061688, then a(n) = (1/n)*Sum_{d|n}mu(d)b(n/d).

Name clarified by Michel Marcus, May 14 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 *)

cp's OEIS Frontend

A091201 Number of orbits of length n under the map whose periodic points are counted by A061688.

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