A301706 -id:A301706 - cp's OEIS frontend

A301595 Number of thrice-partitions of n.

1, 1, 4, 10, 34, 80, 254, 604, 1785, 4370, 11986, 29286, 80355, 193137, 505952, 1239348, 3181970, 7686199, 19520906, 46931241, 117334784, 282021070, 693721166, 1659075192, 4063164983, 9651686516, 23347635094, 55405326513, 133021397071, 313842472333, 749299686508

Offset: 0

Gus Wiseman, Mar 24 2018

nonn

A thrice-partition of n is a choice of a twice-partition of each part in a partition of n. Thrice-partitions correspond to intervals in the lattice form of the multiorder of integer partitions.

			The a(3) = 10 thrice-partitions:
  ((3)), ((21)), ((111)), ((2)(1)), ((11)(1)), ((1)(1)(1)),
  ((2))((1)), ((11))((1)), ((1)(1))((1)),
  ((1))((1))((1)).

Alois P. Heinz, Table of n, a(n) for n = 0..3244
Gus Wiseman, The a(4) = 34 thrice-partitions of 4.

Cf. A000041, A001383, A001970, A061260, A063834, A119442, A196545, A281113, A289501, A300383, A301422, A301462, A301480, A301595, A301598, A301706.

Column k=3 of A323718.

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-> b(n$2, 3):
seq(a(n), n=0..35);  # Alois P. Heinz, Jan 25 2019

twie[n_]:=Sum[Times@@PartitionsP/@ptn,{ptn,IntegerPartitions[n]}];
thrie[n_]:=Sum[Times@@twie/@ptn,{ptn,IntegerPartitions[n]}];
Array[thrie,30]
(* 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_] := b[n, n, 3];
a /@ Range[0, 35] (* Jean-François Alcover, May 19 2021, after Alois P. Heinz *)

O.g.f.: Product_{n > 0} 1/(1 - A063834(n) x^n).

a(0)=1 prepended by Alois P. Heinz, Jan 25 2019

A301764 Number of ways to choose a constant rooted partition of each part in a constant rooted partition of n such that the flattened sequence is also constant.

Original entry on oeis.org

1, 1, 2, 3, 4, 4, 6, 5, 6, 7, 8, 5, 10, 7, 8, 10, 10, 6, 12, 7, 12, 13, 10, 5, 14, 12, 11, 11, 14, 7, 18, 9, 12, 13, 11, 12, 20, 10, 10, 13, 18, 9, 20, 9, 14, 20, 12, 5, 20, 15, 19, 14, 17, 7, 18, 16, 20, 17, 12, 5, 26, 13, 12, 21, 18, 17, 24, 9, 15, 13, 22, 9

Offset: 1

Gus Wiseman, Mar 26 2018

nonn

A rooted partition of n is an integer partition of n - 1.

			The a(11) = 8 rooted twice-partitions: (9), (333), (111111111), (4)(4), (22)(22), (1111)(1111), (1)(1)(1)(1)(1), ()()()()()()()()()().

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A002865, A007425, A063834, A093637, A127524, A295931, A300383, A301422, A301462, A301467, A301480, A301706.

Table[If[n===1,1,DivisorSum[n-1,If[#===1,1,DivisorSigma[0,#-1]]&]],{n,100}]

a(n)=if(n==1, 1, sumdiv(n-1, d, if(d==1, 1, numdiv(d-1)))) \\ Andrew Howroyd, Aug 26 2018

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

Gus Wiseman, Mar 24 2018

nonn

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.

			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)).

Gus Wiseman, The (2*2*3*3) component of the lattice form of the multiorder of integer factorizations.
Gus Wiseman, The (2*2*3*3) component, version 2.

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

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}]

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

A301750 Number of rooted twice-partitions of n where the composite rooted partition is strict.

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 12, 18, 29, 42, 61, 86, 127, 181, 257, 352, 489, 668, 935, 1270, 1730, 2312, 3101, 4112, 5533, 7345, 9742, 12785, 16793, 21821, 28452, 36908, 48108, 62198, 80337, 103081, 132372, 168805, 215247, 273678

Offset: 1

Gus Wiseman, Mar 26 2018

nonn

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.

			The a(8) = 18 rooted twice-partitions where the composite rooted partition is strict:
(6), (51), (42), (321),
(5)(), (41)(), (32)(), (4)(1), (3)(2),
(4)()(), (31)()(), (3)(1)(),
(3)()()(), (21)()()(), (2)(1)()(),
(2)()()()(),
(1)()()()()(),
()()()()()()().

Cf. A002865, A063834, A093637, A127524, A279790, A294788, A301422, A301462, A301467, A301480, A301706.

twirtns[n_]:=Join@@Table[Tuples[IntegerPartitions[#-1]&/@ptn],{ptn,IntegerPartitions[n-1]}];
Table[Select[twirtns[n],UnsameQ@@Join@@#&]//Length,{n,30}]

A301760 Number of rooted twice-partitions of n where the composite rooted partition is constant.

Original entry on oeis.org

1, 1, 2, 4, 7, 11, 17, 24, 34, 46, 63, 82, 109, 140, 183, 233, 298, 376, 479, 598, 753, 938, 1171, 1449, 1797, 2210, 2726, 3342, 4095, 4990, 6088, 7388, 8968, 10843, 13099, 15770, 18975, 22756, 27276, 32603, 38925, 46353, 55158, 65479, 77656, 91904, 108645

Offset: 1

Gus Wiseman, Mar 26 2018

nonn

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.

			The a(5) = 7 rooted twice-partitions: (3), (111), (2)(), (11)(), (1)(1), (1)()(), ()()()().

Cf. A002865, A047968, A063834, A093637, A295924, A301422, A301462, A301467, A301480, A301706.

nn=50;
ser=(1-nn)/(1-x)+Sum[Product[1/(1-x^(d k+1)),{k,0,nn}],{d,nn}];
CoefficientList[Series[ser,{x,0,nn}],x]

O.g.f.: 1/(1 - x) + Sum_{n > 0} (-1/(1 - x) + Product_{k >= 0} 1/(1 - x^(n * k + 1))).

A301766 Number of rooted twice-partitions of n where the first rooted partition is strict and the composite rooted partition is constant, i.e., of type (R,Q,R).

Original entry on oeis.org

1, 1, 1, 3, 4, 6, 7, 9, 11, 13, 16, 19, 22, 26, 32, 36, 42, 52, 59, 66, 79, 93, 108, 125, 141, 162, 192, 222, 248, 285, 331, 375, 430, 492, 555, 632, 719, 816, 929, 1051, 1177, 1327, 1510, 1701, 1908, 2146, 2408, 2705, 3035, 3388, 3792, 4257, 4751, 5284, 5894

Offset: 1

Gus Wiseman, Mar 26 2018

nonn

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.

			The a(9) = 11 rooted twice-partitions:
(7), (1111111),
(6)(), (33)(), (222)(), (111111)(), (11111)(1), (22)(2), (1111)(11),
(1111)(1)(), (111)(11)().

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A002865, A032305, A047966, A063834, A093637, A296134, A300383, A301422, A301462, A301467, A301480, A301706.

twirtns[n_]:=Join@@Table[Tuples[IntegerPartitions[#-1]&/@ptn],{ptn,IntegerPartitions[n-1]}];
Table[Select[twirtns[n],UnsameQ@@Total/@#&&SameQ@@Join@@#&]//Length,{n,20}]

a(n)=if(n<3, 1, sum(k=1, n-2, polcoef(prod(j=0, (n-2)\k, 1 + x^(j*k + 1) + O(x^n)), n-1))) \\ Andrew Howroyd, Aug 26 2018

Terms a(26) and beyond from Andrew Howroyd, Aug 26 2018

A301751 Number of ways to choose a rooted partition of each part in a strict rooted partition of n.

Original entry on oeis.org

1, 1, 1, 3, 5, 10, 17, 32, 54, 100, 166, 289, 494, 840, 1393, 2400, 3931, 6498, 10861, 17728, 28863, 47557, 77042, 123881, 201172, 322459, 517032, 827993, 1316064, 2084632, 3328204, 5236828, 8247676, 13005652, 20417628, 31934709, 49970815, 77789059, 121144373

Offset: 1

Gus Wiseman, Mar 26 2018

nonn

A rooted partition of n is an integer partition of n - 1.

			The a(7) = 17 rooted twice-partitions:
(5), (41), (32), (311), (221), (2111), (11111),
(4)(), (31)(), (22)(), (211)(), (1111)(), (3)(1), (21)(1), (111)(1),
(2)(1)(), (11)(1)().

Andrew Howroyd, Table of n, a(n) for n = 1..500

Cf. A002865, A032305, A063834, A093637, A271619, A281113, A296118, A300383, A301422, A301462, A301467, A301480, A301706.

nn=50;
ser=x*Product[1+PartitionsP[n-1]x^n,{n,nn}];
Table[SeriesCoefficient[ser,{x,0,n}],{n,nn}]

seq(n)={Vec(prod(k=1, n-1, 1 + numbpart(k-1)*x^k + O(x^n)))} \\ Andrew Howroyd, Aug 29 2018

O.g.f.: x * Product_{n > 0} (1 + A000041(n-1) x^n).

A301753 Number of ways to choose a strict rooted partition of each part in a rooted partition of n.

Original entry on oeis.org

1, 1, 2, 3, 6, 9, 16, 25, 43, 66, 108, 166, 269, 408, 643, 975, 1517, 2277, 3497, 5223, 7936, 11803, 17736, 26219, 39174, 57594, 85299, 124957, 183987, 268158, 392685, 569987, 830282, 1200843, 1740422, 2507823, 3620550, 5197885, 7472229, 10694865, 15319700

Offset: 1

Gus Wiseman, Mar 26 2018

nonn

A rooted partition of n is an integer partition of n - 1.

			The a(7) = 16 rooted twice-partitions:
(5), (32), (41),
(2)(2), (3)(1), (4)(), (21)(1), (31)(),
(1)(1)(1), (2)(1)(), (3)()(), (21)()(),
(1)(1)()(), (2)()()(),
(1)()()()(),
()()()()()().

Andrew Howroyd, Table of n, a(n) for n = 1..500

Cf. A002865, A032305, A063834, A093637, A270995, A281113, A296119, A301422, A301462, A301467, A301480, A301706, A301751.

nn=50;
ser=x*Product[1/(1-PartitionsQ[n-1]x^n),{n,nn}];
Table[SeriesCoefficient[ser,{x,0,n}],{n,nn}]

seq(n)={my(u=Vec(prod(k=1, n-1, 1 + x^k + O(x^n)))); Vec(1/prod(k=1, n-1, 1 - u[k]*x^k + O(x^n)))} \\ Andrew Howroyd, Aug 29 2018

O.g.f.: x * Product_{n > 0} 1/(1 - A000009(n-1) x^n).

A301754 Number of ways to choose a strict rooted partition of each part in a strict rooted partition of n.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 8, 13, 18, 29, 44, 67, 100, 150, 217, 326, 470, 690, 1011, 1463, 2099, 3049, 4355, 6214, 8886, 12632, 17885, 25377, 35763, 50252, 70942, 99246, 138600, 193912, 270286, 375471, 522224, 723010, 1000435, 1383002, 1907724, 2624492, 3613885

Offset: 1

Gus Wiseman, Mar 26 2018

nonn

A rooted partition of n is an integer partition of n - 1.

			The a(8) = 13 rooted twice-partitions:
(6), (51), (42), (321),
(5)(), (41)(), (32)(), (4)(1), (31)(1), (3)(2), (21)(2),
(3)(1)(), (21)(1)().

Andrew Howroyd, Table of n, a(n) for n = 1..500

Cf. A002865, A032305, A063834, A093637, A196545, A279785, A296120, A301422, A301462, A301467, A301480, A301706.

nn=50;
ser=x*Product[1+PartitionsQ[n-1]x^n,{n,nn}];
Table[SeriesCoefficient[ser,{x,0,n}],{n,nn}]

seq(n)={my(u=Vec(prod(k=1, n-1, 1 + x^k + O(x^n)))); Vec(prod(k=1, n-1, 1 + u[k]*x^k + O(x^n)))} \\ Andrew Howroyd, Aug 29 2018

O.g.f.: x * Product_{n > 0} (1 + A000009(n-1) x^n).

A301763 Number of ways to choose a constant rooted partition of each part in a constant rooted partition of n.

Original entry on oeis.org

1, 1, 2, 3, 4, 4, 8, 5, 8, 13, 14, 5, 32, 7, 20, 64, 26, 6, 92, 7, 126, 199, 22, 5, 352, 252, 41, 581, 394, 7, 1832, 9, 292, 2119, 31, 3216, 4946, 10, 40, 8413, 7708, 9, 20656, 9, 2324, 53546, 24, 5, 70040, 16395, 59361, 131204, 9503, 7, 266780, 178180, 82086

Offset: 1

Gus Wiseman, Mar 26 2018

nonn

A rooted partition of n is an integer partition of n - 1.

			The a(7) = 8 rooted twice-partitions: (5), (11111), (2)(2), (2)(11), (11)(2), (11)(11), (1)(1)(1), ()()()()()().
The a(15) = 20 rooted twice-partitions:
()()()()()()()()()()()()()(),
(1)(1)(1)(1)(1)(1)(1), (111111)(111111), (1111111111111),
(111111)(222), (222)(111111), (222)(222),
(111111)(33), (222)(33), (33)(111111), (33)(222), (33)(33),
(111111)(6), (222)(6), (33)(6), (6)(111111), (6)(222), (6)(33), (6)(6),
(13).

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A000005, A002865, A047968, A063834, A093637, A127524, A295924, A300383, A301422, A301462, A301467, A301480, A301706.

Table[If[n===1,1,Sum[If[d===n-1,1,DivisorSigma[0,(n-1)/d-1]]^d,{d,Divisors[n-1]}]],{n,50}]

a(n)=if(n==1, 1, sumdiv(n-1, d, if(d==n-1, 1, numdiv((n-1)/d-1)^d))) \\ Andrew Howroyd, Aug 26 2018

cp's OEIS Frontend

A301595 Number of thrice-partitions of n.

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A301764 Number of ways to choose a constant rooted partition of each part in a constant rooted partition of n such that the flattened sequence is also constant.

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A301598 Number of thrice-factorizations of n.

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A301750 Number of rooted twice-partitions of n where the composite rooted partition is strict.

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A301760 Number of rooted twice-partitions of n where the composite rooted partition is constant.

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A301766 Number of rooted twice-partitions of n where the first rooted partition is strict and the composite rooted partition is constant, i.e., of type (R,Q,R).

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A301751 Number of ways to choose a rooted partition of each part in a strict rooted partition of n.

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A301753 Number of ways to choose a strict rooted partition of each part in a rooted partition of n.