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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)).

%I A335078 #23 Nov 10 2020 06:39:09
%S A335078 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,
%T A335078 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,
%U A335078 1,1,1,1,4,4,2,1,1,1,1,1,1,3,3,1,1,1,3,2,1
%N 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)).
%D A335078 Richard Beekman, An Introduction to Number-Theoretic Combinatorics, Lulu Press 2017.
%H A335078 Stefano Spezia, <a href="/A335078/b335078.txt">First 3500 rows of the triangle, flattened</a>
%H A335078 Richard Beekman, <a href="https://www.researchgate.net/publication/341090354_A_General_Solution_of_the_Ferris_Wheel_Problem">A General Solution of the Ferris Wheel Problem</a>, ResearchGate, 2020.
%F A335078 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).
%e A335078 The triangle T(n, k) begins
%e A335078   n\k| 1     2     2     4
%e A335078   ---+--------------------
%e A335078    2 | 1
%e A335078    3 | 1
%e A335078    4 | 1     1
%e A335078    5 | 1
%e A335078    6 | 1     1
%e A335078    7 | 1
%e A335078    8 | 1     1     1
%e A335078    9 | 1     1
%e A335078   10 | 1     1
%e A335078   11 | 1
%e A335078   12 | 1     2     1
%e A335078   13 | 1
%e A335078   14 | 1     1
%e A335078   15 | 1     1
%e A335078   16 | 1     2     1     1
%e A335078   ...
%t A335078 tau[n_,k_]:=If[n==1,1,Product[Binomial[Extract[Extract[FactorInteger[n],i],2]+k,k],{i,Length[FactorInteger[n]]}]]; (* A334997 *)
%t A335078 Nd[n_, m_]:=Sum[(-1)^k*Binomial[m, k]*tau[n, m-k-1], {k,0,m-1}]; (* A334996 *)
%t A335078 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
%o A335078 (PARI) TT(n, k) = if (k==0, 1, sumdiv(n, d, TT(d, k-1))); \\ A334996
%o A335078 U(n, m) = sum(k=0, m-1, (-1)^k*binomial(m, k)*TT(n, m-k-1));
%o A335078 T(n, k) = my(p); (1/k)*sumdiv(k, d, if (ispower(n, d, &p), eulerphi(d)*U(p, k/d)));
%o A335078 row(n) = vector(bigomega(n), k, T(n,k)); \\ _Michel Marcus_, May 25 2020
%Y A335078 Cf. A000012 (1st column), A001222 (Omega), A334996, A334997, A335079 (row sums).
%K A335078 nonn,tabf
%O A335078 2,18
%A A335078 _Stefano Spezia_, May 23 2020