A005804 -id:A005804 - cp's OEIS frontend

A386823 Triangle read by rows: T(n,k) = numerator((n^2 - k^2)/(n^2 + k^2)), where 0 <= k < n.

1, 1, 3, 1, 4, 5, 1, 15, 3, 7, 1, 12, 21, 8, 9, 1, 35, 4, 3, 5, 11, 1, 24, 45, 20, 33, 12, 13, 1, 63, 15, 55, 3, 39, 7, 15, 1, 40, 77, 4, 65, 28, 5, 16, 17, 1, 99, 12, 91, 21, 3, 8, 51, 9, 19, 1, 60, 117, 56, 105, 48, 85, 36, 57, 20, 21, 1, 143, 35, 15, 4, 119, 3, 95, 5, 7, 11, 23

Offset: 1

Stefano Spezia, Aug 04 2025

			The triangle of the fractions begins as:
  1/1;
  1/1,   3/5;
  1/1,   4/5,  5/13;
  1/1, 15/17,   3/5,  7/25;
  1/1, 12/13, 21/29,  8/17,  9/41;
  1/1, 35/37,   4/5,   3/5,  5/13, 11/61;
  1/1, 24/25, 45/53, 20/29, 33/65, 12/37, 13/85;
  ...

Stefano Spezia, First 150 rows of the triangle, flattened

Cf. A000012 (k=0), A000290, A005408, A066830 (k=1), A069011, A094728, A386824 (denominators).

T[n_,k_]:=Numerator[(n^2-k^2)/(n^2+k^2)]; Table[T[n,k],{n,12},{k,0,n-1}]//Flatten

T(n,n-1) = A005804(n-1).

A318120 Number of set partitions of {1,...,n} with relatively prime block sizes.

Original entry on oeis.org

1, 1, 1, 4, 11, 51, 162, 876, 3761, 20782, 109293, 678569, 4038388, 27644436, 186524145, 1379760895, 10323844183, 82864869803, 674798169662, 5832742205056, 51385856585637, 474708148273586, 4486977535287371, 44152005855084345, 444577220573083896

Offset: 0

Gus Wiseman, Dec 16 2018

nonn

			The a(4) = 11 set partitions:
  {{1},{2},{3},{4}}
   {{1},{2},{3,4}}
   {{1},{2,3},{4}}
   {{1},{2,4},{3}}
   {{1,2},{3},{4}}
   {{1,3},{2},{4}}
   {{1,4},{2},{3}}
    {{1},{2,3,4}}
    {{1,2,3},{4}}
    {{1,2,4},{3}}
    {{1,3,4},{2}}

Alois P. Heinz, Table of n, a(n) for n = 0..576

a(n) >= A271426(n).

Cf. A000110, A000258, A000311, A000670, A000740, A000837, A005651, A005804, A008277, A124794, A319182, A320289.

b:= proc(n, t) option remember; `if`(n=0, `if`(t<2, 1, 0),
      add(b(n-j, igcd(t, j))*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..25);  # Alois P. Heinz, Dec 30 2019

numSetPtnsOfType[ptn_]:=Total[ptn]!/Times@@Factorial/@ptn/Times@@Factorial/@Length/@Split[ptn];
Table[Total[numSetPtnsOfType/@Select[IntegerPartitions[n],GCD@@#==1&]],{n,10}]
(* Second program: *)
b[n_, t_] := b[n, t] = If[n == 0, If[t < 2, 1, 0],
     Sum[b[n - j, GCD[t, j]]*Binomial[n - 1, j - 1], {j, 1, n}]];
a[n_] := b[n, 0];
a /@ Range[0, 25] (* Jean-François Alcover, May 10 2021, after Alois P. Heinz *)

a(n) = Sum_{|y| = n, GCD(y) = 1} n! / (Product_i y_i! * Product_i (y)_i!) where the sum is over all relatively prime integer partitions of n and (y)_i is the multiplicity of i in y.

A331685 Number of tree-factorizations of Heinz numbers of integer partitions of n.

Original entry on oeis.org

1, 3, 7, 23, 69, 261, 943, 3815, 15107, 63219, 262791, 1130953, 4838813, 21185125, 92593943, 411160627, 1823656199, 8186105099, 36728532951, 166310761655

Offset: 1

Gus Wiseman, Jan 31 2020

A tree-factorization of n > 1 is either (case 1) the number n itself, or (case 2) a sequence of two or more tree-factorizations, one of each part of a weakly increasing factorization of n into factors > 1.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The a(1) = 1 through a(4) = 23 tree-factorizations:
  2  3      5          7
     4      6          9
     (2*2)  8          10
            (2*3)      12
            (2*4)      16
            (2*2*2)    (2*5)
            (2*(2*2))  (2*6)
                       (2*8)
                       (3*3)
                       (3*4)
                       (4*4)
                       (2*2*3)
                       (2*2*4)
                       (2*2*2*2)
                       (2*(2*3))
                       ((2*2)*4)
                       (2*(2*4))
                       (3*(2*2))
                       (4*(2*2))
                       (2*(2*2*2))
                       (2*2*(2*2))
                       ((2*2)*(2*2))
                       (2*(2*(2*2)))

The orderless version is A319312.
Factorizations are A001055.
P-trees are A196545.
Twice-factorizations are A281113.
Tree-factorizations are A281118.
Enriched p-trees are A289501.
Cf. A000081 A000669, A000720, A001678, A005804, A056239, A112798, A141268, A273873, A330465.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
physemi[n_]:=Prepend[Join@@Table[Tuples[physemi/@f],{f,Select[facs[n],Length[#]>1&]}],n];
Table[Sum[Length[physemi[Times@@Prime/@m]],{m,IntegerPartitions[n]}],{n,8}]

\\ here TF(n) is n terms of A281118 as vector.
TF(n)={my(v=vector(n), w=vector(n)); w[1]=v[1]=1; for(k=2, n, w[k]=v[k]+1; forstep(j=n\k*k, k, -k, my(i=j, e=0); while(i%k==0, i/=k; e++; v[j] += w[k]^e*v[i]))); w}
a(n)={my(v=[prod(i=1, #p, prime(p[i])) | p<-partitions(n)], tf=TF(vecmax(v))); sum(i=1, #v, tf[v[i]])} \\ Andrew Howroyd, Dec 09 2020

a(n) = Sum_i A281118(A215366(n,i)).

a(13)-a(20) from Andrew Howroyd, Dec 09 2020

cp's OEIS Frontend

A386823 Triangle read by rows: T(n,k) = numerator((n^2 - k^2)/(n^2 + k^2)), where 0 <= k < n.

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A318120 Number of set partitions of {1,...,n} with relatively prime block sizes.

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A331685 Number of tree-factorizations of Heinz numbers of integer partitions of n.

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