This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A334996 #44 Aug 21 2025 09:58:11
%S A334996 0,0,1,0,1,0,1,1,0,1,0,1,2,0,1,0,1,2,1,0,1,1,0,1,2,0,1,0,1,4,3,0,1,0,
%T A334996 1,2,0,1,2,0,1,3,3,1,0,1,0,1,4,3,0,1,0,1,4,3,0,1,2,0,1,2,0,1,0,1,6,9,
%U A334996 4,0,1,1,0,1,2,0,1,2,1,0,1,4,3,0,1,0,1,6,6
%N A334996 Irregular triangle read by rows: T(n, m) is the number of ways to distribute Omega(n) objects into precisely m distinct boxes, with no box empty (Omega(n) >= m).
%C A334996 n is the specification number for a set of Omega(n) objects (see Theorem 3 in Beekman's article).
%C A334996 The specification number of a multiset is also called its Heinz number. - _Gus Wiseman_, Aug 25 2020
%C A334996 From _Gus Wiseman_, Aug 25 2020: (Start)
%C A334996 For n > 1, T(n,k) is also the number of ordered factorizations of n into k factors > 1. For example, row n = 24 counts the following ordered factorizations (the first column is empty):
%C A334996 24 3*8 2*2*6 2*2*2*3
%C A334996 4*6 2*3*4 2*2*3*2
%C A334996 6*4 2*4*3 2*3*2*2
%C A334996 8*3 2*6*2 3*2*2*2
%C A334996 12*2 3*2*4
%C A334996 2*12 3*4*2
%C A334996 4*2*3
%C A334996 4*3*2
%C A334996 6*2*2
%C A334996 For n > 1, T(n,k) is also the number of strict length-k chains of divisors from n to 1. For example, row n = 36 counts the following chains (the first column is empty):
%C A334996 36/1 36/2/1 36/4/2/1 36/12/4/2/1
%C A334996 36/3/1 36/6/2/1 36/12/6/2/1
%C A334996 36/4/1 36/6/3/1 36/12/6/3/1
%C A334996 36/6/1 36/9/3/1 36/18/6/2/1
%C A334996 36/9/1 36/12/2/1 36/18/6/3/1
%C A334996 36/12/1 36/12/3/1 36/18/9/3/1
%C A334996 36/18/1 36/12/4/1
%C A334996 36/12/6/1
%C A334996 36/18/2/1
%C A334996 36/18/3/1
%C A334996 36/18/6/1
%C A334996 36/18/9/1
%C A334996 (End)
%D A334996 Richard Beekman, An Introduction to Number-Theoretic Combinatorics, Lulu Press 2017.
%H A334996 Stefano Spezia, <a href="/A334996/b334996.txt">First 3000 rows of the table, flattened</a>
%H A334996 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>.
%F A334996 T(n, m) = Sum_{k=0..m-1} (-1)^k*binomial(m,k)*tau_{m-k-1}(n), where tau_s(r) = A334997(r, s) (see Theorem 3, Lemma 1 and Lemma 2 in Beekman's article).
%F A334996 Conjecture: Sum_{m=0..Omega(n)} T(n, m) = A002033(n-1) for n > 1.
%F A334996 The above conjecture is true since T(n, m) is also the number of ordered factorizations of n into m factors (see Comments) and A002033(n-1) is the number of ordered factorizations of n. - _Stefano Spezia_, Aug 21 2025
%e A334996 The triangle T(n, m) begins
%e A334996 n\m| 0 1 2 3 4
%e A334996 ---+--------------------------
%e A334996 1 | 0
%e A334996 2 | 0 1
%e A334996 3 | 0 1
%e A334996 4 | 0 1 1
%e A334996 5 | 0 1
%e A334996 6 | 0 1 2
%e A334996 7 | 0 1
%e A334996 8 | 0 1 2 1
%e A334996 9 | 0 1 1
%e A334996 10 | 0 1 2
%e A334996 11 | 0 1
%e A334996 12 | 0 1 4 3
%e A334996 13 | 0 1
%e A334996 14 | 0 1 2
%e A334996 15 | 0 1 2
%e A334996 16 | 0 1 3 3 1
%e A334996 ...
%e A334996 From _Gus Wiseman_, Aug 25 2020: (Start)
%e A334996 Row n = 36 counts the following distributions of {1,1,2,2} (the first column is empty):
%e A334996 {1122} {1}{122} {1}{1}{22} {1}{1}{2}{2}
%e A334996 {11}{22} {1}{12}{2} {1}{2}{1}{2}
%e A334996 {112}{2} {11}{2}{2} {1}{2}{2}{1}
%e A334996 {12}{12} {1}{2}{12} {2}{1}{1}{2}
%e A334996 {122}{1} {12}{1}{2} {2}{1}{2}{1}
%e A334996 {2}{112} {1}{22}{1} {2}{2}{1}{1}
%e A334996 {22}{11} {12}{2}{1}
%e A334996 {2}{1}{12}
%e A334996 {2}{11}{2}
%e A334996 {2}{12}{1}
%e A334996 {2}{2}{11}
%e A334996 {22}{1}{1}
%e A334996 (End)
%t A334996 tau[n_,k_]:=If[n==1,1,Product[Binomial[Extract[Extract[FactorInteger[n],i],2]+k,k],{i,1,Length[FactorInteger[n]]}]]; (* A334997 *)
%t A334996 T[n_,m_]:=Sum[(-1)^k*Binomial[m,k]*tau[n,m-k-1],{k,0,m-1}]; Table[T[n,m],{n,1,30},{m,0,PrimeOmega[n]}]//Flatten
%t A334996 (* second program *)
%t A334996 chc[n_]:=If[n==1,{{}},Prepend[Join@@Table[Prepend[#,n]&/@chc[d],{d,DeleteCases[Divisors[n],1|n]}],{n}]]; (* change {{}} to {} if a(1) = 0 *)
%t A334996 Table[Length[Select[chc[n],Length[#]==k&]],{n,30},{k,0,PrimeOmega[n]}] (* _Gus Wiseman_, Aug 25 2020 *)
%o A334996 (PARI) TT(n, k) = if (k==0, 1, sumdiv(n, d, TT(d, k-1))); \\ A334997
%o A334996 T(n, m) = sum(k=0, m-1, (-1)^k*binomial(m, k)*TT(n, m-k-1));
%o A334996 tabf(nn) = {for (n=1, nn, print(vector(bigomega(n)+1, k, T(n, k-1))););} \\ _Michel Marcus_, May 20 2020
%Y A334996 Cf. A000007 (1st column), A000012 (2nd column), A001222 (Omega function), A002033 (row sums shifted left), A007318.
%Y A334996 A008480 gives rows ends.
%Y A334996 A073093 gives row lengths.
%Y A334996 A074206 gives row sums.
%Y A334996 A112798 constructs the multiset with each specification number.
%Y A334996 A124433 is a signed version.
%Y A334996 A251683 is the version with zeros removed.
%Y A334996 A334997 is the non-strict version.
%Y A334996 A337107 is the restriction to factorial numbers.
%Y A334996 A001055 counts factorizations.
%Y A334996 A067824 counts strict chains of divisors starting with n.
%Y A334996 A122651 counts strict chains of divisors summing to n.
%Y A334996 A167865 counts strict chains of divisors > 1 summing to n.
%Y A334996 A253249 counts strict chains of divisors.
%Y A334996 A337105 counts strict chains of divisors from n! to 1.
%Y A334996 Cf. A007425, A008683, A056239, A124010, A167865, A317144, A319001.
%K A334996 nonn,tabf,changed
%O A334996 1,13
%A A334996 _Stefano Spezia_, May 19 2020