cp's OEIS Frontend

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.

Showing 1-4 of 4 results.

A301706 Number of rooted thrice-partitions of n.

Original entry on oeis.org

1, 1, 2, 4, 9, 19, 43, 91, 201, 422, 918, 1896, 4089, 8376, 17793, 36445, 76446, 155209, 324481, 655426, 1355220, 2741092, 5617505, 11291037, 23086423, 46227338, 93753196, 187754647, 378675055, 754695631, 1518414812, 3016719277, 6037006608, 11984729983
Offset: 1

Views

Author

Gus Wiseman, Mar 25 2018

Keywords

Comments

A rooted partition of n is an integer partition of n - 1. A rooted twice-partition of n is a choice of a rooted partition of each part in a rooted partition of n. A rooted thrice-partition of n is a choice of a rooted twice-partition of each part in a rooted partition of n.

Examples

			The a(5) = 9 rooted thrice-partitions:
((2)), ((11)), ((1)()), (()()()),
((1))(), (()())(), (())(()),
(())()(),
()()()().
The a(6) = 19 rooted thrice-partitions:
((3)), ((21)), ((111)), ((2)()), ((11)()), ((1)(1)), ((1)()()), (()()()()),
((2))(), ((11))(), ((1)())(), (()()())(), ((1))(()), (()())(()),
((1))()(), (()())()(), (())(())(),
(())()()(),
()()()()().

Crossrefs

Cf. A000041, A001383, A002865, A063834, A093637, A119442, A196545, A281113, A289501, A300383, A301422, A301462, A301467, A301480, A301595, A301598.

Programs

  • Mathematica
    twire[n_]:=twire[n]=Sum[Times@@PartitionsP/@(ptn-1),{ptn,IntegerPartitions[n-1]}];
thrire[n_]:=Sum[Times@@twire/@ptn,{ptn,IntegerPartitions[n-1]}];
Array[thrire,30]

A323718 Array read by antidiagonals upwards where A(n,k) is the number of k-times partitions of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 1, 1, 1, 5, 6, 4, 1, 1, 1, 7, 15, 10, 5, 1, 1, 1, 11, 28, 34, 15, 6, 1, 1, 1, 15, 66, 80, 65, 21, 7, 1, 1, 1, 22, 122, 254, 185, 111, 28, 8, 1, 1, 1, 30, 266, 604, 739, 371, 175, 36, 9, 1, 1, 1, 42, 503, 1785, 2163, 1785, 672, 260, 45, 10, 1, 1
Offset: 0

Views

Author

Gus Wiseman, Jan 25 2019

Keywords

Comments

A k-times partition of n for k > 1 is a sequence of (k-1)-times partitions, one of each part in an integer partition of n. A 1-times partition of n is just an integer partition of n, and the only 0-times partition of n is the number n itself.

Examples

			Array begins:
       k=0:   k=1:   k=2:   k=3:   k=4:   k=5:
  n=0:  1      1      1      1      1      1
  n=1:  1      1      1      1      1      1
  n=2:  1      2      3      4      5      6
  n=3:  1      3      6     10     15     21
  n=4:  1      5     15     34     65    111
  n=5:  1      7     28     80    185    371
  n=6:  1     11     66    254    739   1785
  n=7:  1     15    122    604   2163   6223
  n=8:  1     22    266   1785   8120  28413
  n=9:  1     30    503   4370  24446 101534
The A(4,2) = 15 twice-partitions:
  (4)  (31)    (22)    (211)      (1111)
       (3)(1)  (2)(2)  (11)(2)    (11)(11)
                       (2)(11)    (111)(1)
                       (21)(1)    (11)(1)(1)
                       (2)(1)(1)  (1)(1)(1)(1)

Links

Crossrefs

Columns: A000012 (k=0), A000041 (k=1), A063834 (k=2), A301595 (k=3).
Rows: A000027 (n=2), A000217 (n=3), A006003 (n=4).
Main diagonal gives A306187.
Cf. A001970, A055884, A096751, A144150, A196545, A281113, A289501, A290353, A300383, A323719, A327618, A327639.

Programs

  • Maple
    b:= proc(n, i, k) option remember; `if`(n=0 or k=0 or i=1,
      1, b(n, i-1, k)+b(i$2, k-1)*b(n-i, min(n-i, i), k))
    end:
A:= (n, k)-> b(n$2, k):
seq(seq(A(d-k, k), k=0..d), d=0..14);  # Alois P. Heinz, Jan 25 2019
  • Mathematica
    ptnlev[n_,k_]:=Switch[k,0,{n},1,IntegerPartitions[n],_,Join@@Table[Tuples[ptnlev[#,k-1]&/@ptn],{ptn,IntegerPartitions[n]}]];
Table[Length[ptnlev[sum-k,k]],{sum,0,12},{k,0,sum}]
(* Second program: *)
b[n_, i_, k_] := b[n, i, k] = If[n == 0 || k == 0 || i == 1, 1,
     b[n, i - 1, k] + b[i, i, k - 1]*b[n - i, Min[n - i, i], k]];
A[n_, k_] := b[n, n, k];
Table[Table[A[d - k, k], {k, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, May 13 2021, after Alois P. Heinz *)

Formula

Column k is the formal power product transform of column k-1, where the formal power product transform of a sequence q with offset 1 is the sequence whose ordinary generating function is Product_{n >= 1} 1/(1 - q(n) * x^n).
A(n,k) = Sum_{i=0..k} binomial(k,i) * A327639(n,i). - Alois P. Heinz, Sep 20 2019

A301598 Number of thrice-factorizations of n.

Original entry on oeis.org

1, 1, 1, 4, 1, 4, 1, 10, 4, 4, 1, 16, 1, 4, 4, 34, 1, 16, 1, 16, 4, 4, 1, 54, 4, 4, 10, 16, 1, 22, 1, 80, 4, 4, 4, 78, 1, 4, 4, 54, 1, 22, 1, 16, 16, 4, 1, 181, 4, 16, 4, 16, 1, 54, 4, 54, 4, 4, 1, 102, 1, 4, 16, 254, 4, 22, 1, 16, 4, 22, 1, 272, 1, 4, 16, 16
Offset: 1

Views

Author

Gus Wiseman, Mar 24 2018

Keywords

Comments

A thrice-factorization of n is a choice of a twice-factorization of each factor in a factorization of n. Thrice-factorizations correspond to intervals in the lattice form of the multiorder of integer factorizations.

Examples

			The a(12) = 16 thrice-factorizations:
((2))*((2))*((3)), ((2))*((2)*(3)), ((3))*((2)*(2)), ((2)*(2)*(3)),
((2))*((2*3)), ((2)*(2*3)),
((2))*((6)), ((2)*(6)),
((3))*((2*2)), ((3)*(2*2)),
((3))*((4)), ((3)*(4)),
((2*2*3)),
((2*6)),
((3*4)),
((12)).

Links

Crossrefs

Cf. A001055, A007716, A050336, A050338, A063834, A162247, A269134, A281113, A281116, A301595, A301598, A301706.

Programs

  • Mathematica
    facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
twifacs[n_]:=Join@@Table[Tuples[facs/@f],{f,facs[n]}];
thrifacs[n_]:=Join@@Table[Tuples[twifacs/@f],{f,facs[n]}];
Table[Length[thrifacs[n]],{n,15}]

Formula

Dirichlet g.f.: Product_{n > 1} 1/(1 - A281113(n)/n^s).

A324930 Total weight of the multiset of multisets of multisets with MMM number n. Totally additive with a(prime(n)) = A302242(n).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 2, 0, 0, 1, 0, 1, 2, 0, 2, 1, 0, 0, 1, 1, 1, 0, 1, 2, 1, 0, 1, 0, 1, 1, 2, 0, 2, 1, 1, 2, 2, 0, 0, 2, 2, 1, 0, 0, 2, 0, 0, 1, 1, 1, 2, 1, 0, 0, 2, 1, 3, 2, 2, 1, 1, 0, 3, 1, 2, 0, 1, 1, 1, 1, 0, 2, 2, 0, 3, 2, 1
Offset: 1

Views

Author

Gus Wiseman, Mar 21 2019

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The finite multiset of finite multisets of finite multisets of positive integers with MMM number n is obtained by factoring n into prime numbers, then factoring each of their prime indices into prime numbers, then factoring each of their prime indices into prime numbers, and finally taking their prime indices.

Examples

			The sequence of all finite multisets of finite multisets of finite multisets of positive integers begins (o is the empty multiset):
   1: o
   2: (o)
   3: ((o))
   4: (oo)
   5: (((1)))
   6: (o(o))
   7: ((oo))
   8: (ooo)
   9: ((o)(o))
  10: (o((1)))
  11: (((2)))
  12: (oo(o))
  13: ((o(1)))
  14: (o(oo))
  15: ((o)((1)))
  16: (oooo)
  17: (((11)))
  18: (o(o)(o))
  19: ((ooo))
  20: (oo((1)))

Crossrefs

Cf. A000081, A000720, A001222, A050338, A056239, A112798, A301595, A302242, A318564, A318565, A318566, A324928.

Programs

  • Mathematica
    fi[n_]:=If[n==1,{},FactorInteger[n]];
Table[Total[Cases[fi[n],{p_,k_}:>k*Total[Cases[fi[PrimePi[p]],{q_,j_}:>j*PrimeOmega[PrimePi[q]]]]]],{n,60}]
Showing 1-4 of 4 results.

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