A301462 -id:A301462 - cp's OEIS frontend

A047966 a(n) = Sum_{ d divides n } q(d), where q(d) = A000009 = number of partitions of d into distinct parts.

1, 2, 3, 4, 4, 8, 6, 10, 11, 15, 13, 25, 19, 29, 33, 42, 39, 62, 55, 81, 84, 103, 105, 153, 146, 185, 203, 253, 257, 344, 341, 432, 463, 552, 594, 747, 761, 920, 1003, 1200, 1261, 1537, 1611, 1921, 2089, 2410, 2591, 3095, 3270, 3815, 4138, 4769, 5121, 5972, 6394, 7367, 7974, 9066, 9793, 11305, 12077, 13736, 14940

Offset: 1

N. J. A. Sloane

nonn

Number of partitions of n such that every part occurs with the same multiplicity. - Vladeta Jovovic, Oct 22 2004
Christopher and Christober call such partitions uniform. - Gus Wiseman, Apr 16 2018
Equals inverse Mobius transform (A051731) * A000009, where the latter begins (1, 1, 2, 2, 3, 4, 5, 6, 8, ...). -  Gary W. Adamson, Jun 08 2009

			The a(6) = 8 uniform partitions are (6), (51), (42), (33), (321), (222), (2211), (111111). - _Gus Wiseman_, Apr 16 2018

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
A. David Christopher and M. Davamani Christober, Relatively Prime Uniform Partitions, Gen. Math. Notes, Vol. 13, No. 2, December, 2012, pp.1-12.

Cf. A000009, A000837, A024994, A047968, A063834, A279788, A289501, A300383, A301462, A302698.

with(numtheory):
b:= proc(n) option remember; `if`(n=0, 1, add(add(
     `if`(d::odd, d, 0), d=divisors(j))*b(n-j), j=1..n)/n)
    end:
a:= n-> add(b(d), d=divisors(n)):
seq(a(n), n=1..100);  # Alois P. Heinz, Jul 11 2016

b[n_] := b[n] = If[n==0, 1, Sum[DivisorSum[j, If[OddQ[#], #, 0]&]*b[n-j], {j, 1, n}]/n]; a[n_] := DivisorSum[n, b]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Dec 06 2016 after Alois P. Heinz *)
Table[DivisorSum[n,PartitionsQ],{n,20}] (* Gus Wiseman, Apr 16 2018 *)

N = 66; q='q+O('q^N);
D(q)=eta(q^2)/eta(q); \\ A000009
Vec( sum(e=1,N,D(q^e)-1) ) \\ Joerg Arndt, Mar 27 2014

G.f.: Sum_{k>0} (-1+Product_{i>0} (1+z^(k*i))). - Vladeta Jovovic, Jun 22 2003
G.f.: Sum_{k>=1} q(k)*x^k/(1 - x^k), where q() = A000009. - Ilya Gutkovskiy, Jun 20 2018
a(n) ~ exp(Pi*sqrt(n/3)) / (4*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Aug 27 2018

A301467 Number of enriched r-trees of size n with no empty subtrees.

Original entry on oeis.org

1, 2, 4, 8, 20, 48, 136, 360, 1040, 2944, 8704, 25280, 76320, 226720, 692992, 2096640, 6470016, 19799936, 61713152, 190683520, 598033152, 1863995392, 5879859200, 18438913536, 58464724992, 184356152832, 586898946048, 1859875518464, 5941384080384, 18901502482432

Offset: 1

Gus Wiseman, Mar 21 2018

nonn

An enriched r-tree of size n > 0 with no empty subtrees is either a single node of size n, or a finite nonempty sequence of enriched r-trees with no empty subtrees and with weakly decreasing sizes summing to n - 1.

			The a(4) = 8 enriched r-trees with no empty subtrees: 4, (3), (21), ((2)), (111), ((11)), ((1)1), (((1))).
The a(5) = 20 enriched r-trees with no empty subtrees:
  5,
  (4), ((3)), ((21)), (((2))), ((111)), (((11))), (((1)1)), ((((1)))),
  (31), (22), (2(1)), ((2)1), ((1)2), ((11)1), ((1)(1)), (((1))1),
  (211), ((1)11),
  (1111).

Alois P. Heinz, Table of n, a(n) for n = 1..1910

Cf. A000081, A004111, A032305, A055277, A093637, A127524, A196545, A289501, A300660, A301342-A301345, A301364-A301368, A301422, A301462, A301469, A301470.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1)* a(i)^j, j=0..n/i)))
    end:
a:= n-> `if`(n<2, n, 1+b(n-1$2)):
seq(a(n), n=1..30);  # Alois P. Heinz, Jun 21 2018

pert[n_]:=pert[n]=If[n===1,1,1+Sum[Times@@pert/@y,{y,IntegerPartitions[n-1]}]];
Array[pert,30]
(* Second program: *)
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0,
     Sum[b[n - i*j, i - 1] a[i]^j, {j, 0, n/i}]]];
a[n_] := a[n] = If[n < 2, n, 1 + b[n-1, n-1]];
Array[a, 30] (* Jean-François Alcover, May 09 2021, after Alois P. Heinz *)

seq(n)={my(v=vector(n)); v[1]=1; for(n=2, n, v[n] = 1 + polcoef(1/prod(k=1, n-1, 1 - v[k]*x^k + O(x^n)), n-1)); v} \\ Andrew Howroyd, Aug 26 2018

O.g.f.: x^2/(1 - x) + x Product_{i > 0} 1/(1 - a(i) x^i).

A301480 Number of rooted twice-partitions of n.

Original entry on oeis.org

1, 1, 2, 4, 8, 15, 30, 54, 103, 186, 345, 606, 1115, 1936, 3466, 6046, 10630, 18257, 31927, 54393, 93894, 159631, 272155, 458891, 779375, 1305801, 2196009, 3667242, 6130066, 10170745, 16923127, 27942148, 46211977, 76039205, 125094369, 204952168, 335924597

Offset: 1

Gus Wiseman, Mar 22 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) = 8 rooted twice-partitions: ((3)), ((21)), ((111)), ((2)()), ((11)()), ((1)(1)), ((1)()()), (()()()()).
The a(6) = 15 rooted twice-partitions:
(4), (31), (22), (211), (1111),
(3)(), (21)(), (111)(), (2)(1), (11)(1),
(2)()(), (11)()(), (1)(1)(),
(1)()()(),
()()()()().

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

Cf. A001383, A002865, A032305, A063834, A093637, A127524, A196545, A220418, A281113, A289501, A301422, A301462, A301467.

nn=30;
ser=x*Product[1/(1-PartitionsP[n-1]x^n),{n,nn}];
Table[SeriesCoefficient[ser,{x,0,n}],{n,nn}]

seq(n)={Vec(1/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/(1 - P(n-1) x^n) where P = A000041.

A300486 Number of relatively prime or monic partitions of n.

Original entry on oeis.org

1, 2, 3, 4, 7, 8, 15, 18, 28, 35, 56, 64, 101, 120, 168, 210, 297, 348, 490, 583, 776, 946, 1255, 1482, 1952, 2335, 2981, 3581, 4565, 5387, 6842, 8119, 10086, 12013, 14863, 17527, 21637, 25525, 31083, 36695, 44583, 52256, 63261, 74171, 88932, 104303, 124754

Offset: 1

Gus Wiseman, Apr 15 2018

nonn

A relatively prime or monic partition of n is an integer partition of n that is either of length 1 (monic) or whose parts have no common divisor other than 1 (relatively prime).

			The a(6) = 8 relatively prime or monic partitions are (6), (51), (411), (321), (3111), (2211), (21111), (111111). Missing from this list are (42), (33), (222).

Andrew Howroyd, Table of n, a(n) for n = 1..1000
A. David Christopher and M. Davamani Christober, Relatively Prime Uniform Partitions, Gen. Math. Notes, Vol. 13, No. 2, December, 2012, pp. 1-12.

Cf. A000837, A001383, A063834, A093637, A196545, A281113, A289501, A300383, A301462, A301467, A301480, A302094, A302698, A302915, A302916, A302917.

Table[Length[Select[IntegerPartitions[n],Or[Length[#]===1,GCD@@#===1]&]],{n,20}]

a(n)={(n > 1) + sumdiv(n, d, moebius(d)*numbpart(n/d))} \\ Andrew Howroyd, Aug 29 2018

a(n > 1) = 1 + A000837(n) = 1 + Sum_{d|n} mu(d) * A000041(n/d).

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

Gus Wiseman, Mar 25 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. A rooted thrice-partition of n is a choice of a rooted twice-partition of each part in a rooted partition of n.

			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))()(), (()())()(), (())(())(),
(())()()(),
()()()()().

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

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]

A301595 Number of thrice-partitions of n.

Original entry on oeis.org

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

A301470 Signed recurrence over enriched r-trees: a(n) = (-1)^n + Sum_y Product_{i in y} a(y) where the sum is over all integer partitions of n - 1.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 2, 3, 5, 9, 15, 27, 47, 87, 155, 288, 524, 983, 1813, 3434, 6396, 12174, 22891, 43810, 82925, 159432, 303559, 585966, 1121446, 2171341, 4172932, 8106485, 15635332, 30445899, 58925280, 115014681, 223210718, 436603718, 849480835, 1664740873

Offset: 0

Gus Wiseman, Mar 21 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..3266

Cf. A032305, A055277, A093637, A127524, A196545, A220418, A273866, A273873, A289501, A290261, A290971, A301342-A301345, A301364-A301368, A301422, A301462, A301467, A301469.

b:= proc(n, i) option remember; `if`(n=0, 1,
     `if`(i<1, 0, b(n, i-1)+a(i)*b(n-i, min(n-i, i))))
    end:
a:= n-> `if`(n<2, 1-n, b(n-2$2)+b(n-1, n-2)):
seq(a(n), n=0..45);  # Alois P. Heinz, Jun 23 2018

a[n_]:=a[n]=(-1)^n+Sum[Times@@a/@y,{y,IntegerPartitions[n-1]}];
Array[a,30]
(* Second program: *)
b[n_, i_] := b[n, i] = If[n == 0, 1,
     If[i < 1, 0, b[n, i - 1] + a[i] b[n - i, Min[n - i, i]]]];
a[n_] := If[n < 2, 1 - n, b[n - 2, n - 2] + b[n - 1, n - 2]];
a /@ Range[0, 45] (* Jean-François Alcover, May 20 2021, after Alois P. Heinz *)

O.g.f.: 1/(1 + x) + x Product_{i > 0} 1/(1 - a(i) x^i).

a(n) = Sum_t (-1)^w(t) where the sum is over all enriched r-trees of size n and w(t) is the sum of leaves of t.

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

A302094 Number of relatively prime or monic twice-partitions of n.

Original entry on oeis.org

1, 3, 6, 10, 27, 35, 113, 170, 396, 641, 1649, 2318, 5905, 9112, 18678, 32529, 69094, 106210, 227480, 363433, 705210, 1196190, 2325023, 3724233, 7192245, 11915884, 21857887, 36597843, 67406158, 109594872, 201747847, 333400746, 591125465, 987069077, 1743223350

Offset: 1

Gus Wiseman, Apr 15 2018

nonn

A relatively prime or monic partition of n is an integer partition of n that is either of length 1 (monic) or whose parts have no common divisor other than 1 (relatively prime). Then a relatively prime or monic twice-partition of n is a choice of a relatively prime or monic partition of each part in a relatively prime or monic partition of n.

			The a(4) = 10 relatively prime or monic twice-partitions:
(4), (31), (211), (1111),
(3)(1), (21)(1), (111)(1),
(2)(1)(1), (11)(1)(1),
(1)(1)(1)(1).

A. David Christopher and M. Davamani Christober, Relatively Prime Uniform Partitions, Gen. Math. Notes, Vol. 13, No. 2, December, 2012, pp. 1-12.

Cf. A000837, A001383, A063834, A093637, A196545, A281113, A289501, A300383, A300486, A301462, A301467, A301480, A302915, A302916, A302917.

ip[n_]:=ip[n]=Select[IntegerPartitions[n],Or[Length[#]===1,GCD@@#===1]&];
Table[Sum[Times@@Length/@ip/@ptn,{ptn,ip[n]}],{n,10}]

A302915 Number of relatively prime enriched p-trees of weight n.

Original entry on oeis.org

1, 2, 4, 8, 28, 56, 256, 656, 2480, 6688, 30736, 73984, 366560, 1006720, 3966976, 12738560, 58427648, 148069632, 764473600, 2133585664, 8939502080, 28705390592, 136987259648, 356634376704, 1780025034240, 5455065263104, 23215437079552, 73123382895616

Offset: 1

Gus Wiseman, Apr 15 2018

nonn

A relatively prime enriched p-tree of weight n is either a single node of weight n, or a finite sequence of two or more relatively prime enriched p-trees whose weights are weakly decreasing, relatively prime, and sum to n.

			The a(4) = 8 relatively prime enriched p-trees are 4, (31), ((21)1), (((11)1)1), ((111)1), (211), ((11)11), (1111). Missing from this list are the enriched p-trees ((11)(11)), ((11)2), (2(11)), (22).

Cf. A000081, A000837, A003238, A004111, A055277, A093637, A196545, A289501, A290689, A300486, A301462, A301467, A302094, A302916, A302917.

a[n_]:=a[n]=1+Sum[Times@@a/@y,{y,Rest[Select[IntegerPartitions[n],Or[Length[#]===1,GCD@@#===1]&]]}];
Array[a,20]

cp's OEIS Frontend

A047966 a(n) = Sum_{ d divides n } q(d), where q(d) = A000009 = number of partitions of d into distinct parts.

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A301467 Number of enriched r-trees of size n with no empty subtrees.

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A301480 Number of rooted twice-partitions of n.

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A300486 Number of relatively prime or monic partitions of n.

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A301706 Number of rooted thrice-partitions of n.

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A301595 Number of thrice-partitions of n.

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A301470 Signed recurrence over enriched r-trees: a(n) = (-1)^n + Sum_y Product_{i in y} a(y) where the sum is over all integer partitions of n - 1.

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