A061693 -id:A061693 - cp's OEIS frontend

A092239 Number of orbits of length n under the map whose periodic points are counted by A061693.

0, 2, 9, 42, 225, 1260, 7497, 46176, 293382, 1908150, 12655269, 85287870, 582628683, 4026368514, 28104231825, 197884340160, 1404038987577, 10029929788566, 72086075552493, 520920674929650

Offset: 1

Thomas Ward, Feb 24 2004

nonn

Old name was: A061693 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.

			a(3)=9 since a(3)=(1/3)(b(3)-b(1)) where b is the sequence A061693, which starts 0,4,27.

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. A061693.

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 *)

If b(n) is the n-th term of A061693, then a(n) = (1/n)*Sum_{d|n}mu(d)a(n/d).

a(n) ~ 8^n / (Pi*sqrt(3)*n^2). - Vaclav Kotesovec, Sep 05 2019

Name clarified by Michel Marcus, May 14 2015

A061692 Triangle of generalized Stirling numbers.

Original entry on oeis.org

1, 1, 4, 1, 27, 36, 1, 172, 864, 576, 1, 1125, 17500, 36000, 14400, 1, 7591, 351000, 1746000, 1944000, 518400, 1, 52479, 7197169, 80262000, 191394000, 133358400, 25401600, 1, 369580, 151633440, 3691514176, 17188416000, 23866214400, 11379916800, 1625702400

Offset: 1

N. J. A. Sloane, Jun 19 2001

			1; 1,4; 1,27,36; 1,172,864,576; ...

Alois P. Heinz, Rows n = 1..100, 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.

Diagonals give A001044, A061695, A061693, A061694. Cf. A061691.

Row sums give A061684.

b:= proc(n) option remember; expand(
      `if`(n=0, 1, add(x*b(n-i)/i!^3, i=1..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i)/i!, i=1..n))(b(n)*n!^3):
seq(T(n), n=1..10);  # Alois P. Heinz, Sep 10 2019

R[0, ] = 1; R[n, x_] := R[n, x] = x*Sum[Binomial[n, k]^2*Binomial[n-1, k]*R[k, x], {k, 0, n-1}]; Table[CoefficientList[R[n, x], x] // Rest, {n, 1, 8}] // Flatten (* Jean-François Alcover, Sep 01 2015, after Peter Bala *)

T(n, k) = 1/k!*Sum multinomial(n, n_1, n_2, ..n_k)^3, where the sum extends over all compositions (n_1, n_2, .., n_k) of n into exactly k nonnegative parts. - Vladeta Jovovic, Apr 23 2003

The row polynomials R(n,x) satisfy the recurrence equation R(n,x) = x*( sum {k = 0..n-1} binomial(n,k)^2*binomial(n-1,k)*R(k,x) ) with R(0,x) = 1. Also R(n,x + y) = sum {k = 0..n} binomial(n,k)^3*R(k,x)*R(n-k,y). - Peter Bala, Sep 17 2013

More terms from Vladeta Jovovic, Apr 23 2003

cp's OEIS Frontend

A092239 Number of orbits of length n under the map whose periodic points are counted by A061693.

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