A335078 Irregular triangle read by rows: T(n, k) is the number of rotationally inequivalent Ferris Wheel distributions of Omega(n) colored balls of specification number n into precisely k boxes of the Ferris Wheel, with no box empty (0 < k <= Omega(n)).

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 4, 2, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 3, 2, 1

Offset

2,18

References

Richard Beekman, An Introduction to Number-Theoretic Combinatorics, Lulu Press 2017.

Links

Stefano Spezia, First 3500 rows of the triangle, flattened

Richard Beekman, A General Solution of the Ferris Wheel Problem, ResearchGate, 2020.

Formula

T(n, k) = (1/k)*Sum_{d divides k and n^(1/d) is a positive integer} phi(d)*A334996(n^(1/d), k/d) (see Theorem 4 in Beekman's article).

Example

The triangle T(n, k) begins
  n\k| 1     2     2     4
  ---+--------------------
   2 | 1
   3 | 1
   4 | 1     1
   5 | 1
   6 | 1     1
   7 | 1
   8 | 1     1     1
   9 | 1     1
  10 | 1     1
  11 | 1
  12 | 1     2     1
  13 | 1
  14 | 1     1
  15 | 1     1
  16 | 1     2     1     1
  ...

Mathematica

tau[n_, k_]:=If[n==1, 1, Product[Binomial[Extract[Extract[FactorInteger[n], i], 2]+k, k], {i, Length[FactorInteger[n]]}]]; (* A334997 *)
Nd[n_, m_]:=Sum[(-1)^k*Binomial[m, k]*tau[n, m-k-1], {k, 0, m-1}]; (* A334996 *)
T[n_, k_]:=1/k*DivisorSum[k, EulerPhi[#]*Nd[n^(1/#), k/#]&, IntegerQ[n^(1/#)]&];  Table[T[n, m], {n, 2, 43}, {m, PrimeOmega[n]}]//Flatten

Prog

(PARI) TT(n, k) = if (k==0, 1, sumdiv(n, d, TT(d, k-1))); \\ A334996
U(n, m) = sum(k=0, m-1, (-1)^k*binomial(m, k)*TT(n, m-k-1));
T(n, k) = my(p); (1/k)*sumdiv(k, d, if (ispower(n, d, &p), eulerphi(d)*U(p, k/d)));
row(n) = vector(bigomega(n), k, T(n, k)); \\ Michel Marcus, May 25 2020

Crossrefs

Cf. A000012 (1st column), A001222 (Omega), A334996, A334997, A335079 (row sums).

Sequence in context: A163379 A006466 A316439 * A086597 A322320 A369008

Adjacent sequences: A335075 A335076 A335077 * A335079 A335080 A335081

Keyword

nonn,tabf

Author

Stefano Spezia, May 23 2020

Status

approved