A289501 -id:A289501 - cp's OEIS frontend

A331875 Number of enriched identity p-trees of weight n.

1, 1, 2, 3, 6, 14, 32, 79, 198, 522, 1368, 3716, 9992, 27612, 75692, 212045, 589478, 1668630, 4690792, 13387332, 37980664, 109098556, 311717768, 900846484, 2589449032, 7515759012, 21720369476, 63305262126, 183726039404, 537364221200, 1565570459800, 4592892152163

Offset: 1

Gus Wiseman, Jan 31 2020

nonn

An enriched identity p-tree of weight n is either the number n itself or a finite sequence of distinct enriched identity p-trees whose weights are weakly decreasing and sum to n.

			The a(1) = 1 through a(6) = 14 enriched p-trees:
  1  2  3     4        5           6
        (21)  (31)     (32)        (42)
              ((21)1)  (41)        (51)
                       ((21)2)     (321)
                       ((31)1)     ((21)3)
                       (((21)1)1)  ((31)2)
                                   ((32)1)
                                   (3(21))
                                   ((41)1)
                                   ((21)21)
                                   (((21)1)2)
                                   (((21)2)1)
                                   (((31)1)1)
                                   ((((21)1)1)1)

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

The orderless version is A300660.
The locally disjoint case is A331684.
Identity trees are A004111.
P-trees are A196545.
Enriched p-trees are A289501.
Cf. A000669, A141268, A306200, A316471, A331683, A331685, A331686, A331783.

eptrid[n_]:=Prepend[Select[Join@@Table[Tuples[eptrid/@p],{p,Rest[IntegerPartitions[n]]}],UnsameQ@@#&],n];
Table[Length[eptrid[n]],{n,10}]

seq(n)={my(v=vector(n)); for(n=1, n, v[n] = 1 + polcoef(prod(k=1, n-1, sum(j=0, n\k, j!*binomial(v[k],j)*x^(k*j)) + O(x*x^n)), n)); v} \\ Andrew Howroyd, Feb 09 2020

Terms a(21) and beyond from Andrew Howroyd, Feb 09 2020

A290971 Write x/(1-x) in the form Sum_{j>=1} a(j)x^j/(1+a(j)x^j).

Original entry on oeis.org

1, 2, 0, 6, 0, -6, 0, 54, 0, -30, 0, -114, 0, -126, 0, 4470, 0, -294, 0, -5850, 0, -2046, 0, -92418, 0, -8190, 0, -247674, 0, 2010, 0, 30229110, 0, -131070, 0, -8200914, 0, -524286, 0, -362617770, 0, 183162, 0, -354416634, 0, -8388606, 0, -53614489794, 0

Offset: 1

Gus Wiseman, Aug 16 2017

sign

Cf. A048272, A220418, A260685, A273873, A275870, A281145, A289078, A289501, A290261, A290973.

nn=20;-Solve[Table[Sum[a[n/d]^d,{d,Divisors[n]}]==-1,{n,nn}],Array[a,nn]][[1,All,2]]

a(n) = -Sum_t (-1)^v(t) where the sum is over all same-trees of weight n (see A281145 for definition) and v(t) is the number of nodes (branchings and leaves) in t.

A300352 Number of strict trees of weight n with distinct leaves.

Original entry on oeis.org

1, 1, 2, 2, 3, 6, 8, 11, 17, 40, 48, 76, 109, 159, 400, 470, 745, 1057, 1576, 2103, 5267, 6022, 9746, 13390, 20099, 26542, 39396, 82074, 101387, 152291, 215676, 308937, 423587, 596511, 799022, 1623311, 1960223, 2947722, 4048704, 5845982, 7794809, 11028888

Offset: 1

Gus Wiseman, Mar 03 2018

nonn

A strict tree of weight n > 0 is either a single node of weight n, or a sequence of two or more strict trees with strictly decreasing weights summing to n.

			The a(8) = 11 strict trees with distinct leaves: 8, (71), ((52)1), ((43)1), (62), ((51)2), (53), ((41)3), (5(21)), (521), (431).

Cf. A000009, A056239, A063834, A196545, A246867, A273873, A281145, A289501, A294018, A294079, A296150, A299201, A299202, A299203, A300353, A300354, A300355.

sps[{}]:={{}};sps[set:{i_,_}]:=
Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
str[q_]:=str[q]=If[Length[q]===1,1,Total[Times@@@Map[str,Select[sps[q],And[Length[#]>1,UnsameQ@@Total/@#]&],{2}]]];
Table[Total[str/@Select[IntegerPartitions[n],UnsameQ@@#&]],{n,1,20}]

a(n) = Sum_{i=1..A000009(n)} A294018(A246867(n,i)).

A300442 Number of binary strict trees of weight n.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 10, 23, 46, 108, 231, 561, 1285, 3139, 7348, 18265, 43907, 109887, 267582, 675866, 1669909, 4238462, 10555192, 26955062, 67706032, 173591181, 438555624, 1129088048, 2869732770, 7410059898, 18911818801, 48986728672, 125562853003, 326011708368

Offset: 0

Gus Wiseman, Mar 05 2018

nonn

A binary strict tree of weight n > 0 is either a single node of weight n, or an ordered pair of binary strict trees with strictly decreasing weights summing to n.

			The a(5) = 6 binary strict trees: 5, (41), (32), ((31)1), ((21)2), (((21)1)1).
The a(6) = 10 binary strict trees:
  6,
  (51), (42),
  ((41)1), ((32)1), ((31)2),
  (((31)1)1), (((21)2)1), (((21)1)2),
  ((((21)1)1)1).

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000992, A001190, A063834, A196545, A273873, A289501, A292432, A293511, A300352, A300440, A300443.

a:= proc(n) option remember;
      1+add(a(j)*a(n-j), j=1..(n-1)/2)
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Mar 06 2018

k[n_]:=k[n]=1+Sum[Times@@k/@y,{y,Select[IntegerPartitions[n],Length[#]===2&&UnsameQ@@#&]}];
Array[k,40]
(* Second program: *)
a[n_] := a[n] = 1 + Sum[a[j]*a[n - j], {j, 1, (n - 1)/2}];
a /@ Range[0, 40] (* Jean-François Alcover, May 13 2021, after Alois P. Heinz *)

seq(n)={my(v=vector(n)); for(n=1, n, v[n] = 1 + sum(k=1, (n-1)\2, v[k]*v[n-k])); concat([1], v)} \\ Andrew Howroyd, Aug 25 2018

a(n) = 1 + Sum_{x + y = n, 0 < x < y < n} a(x) * a(y).

A301368 Regular triangle where T(n,k) is the number of binary enriched p-trees of weight n with k leaves.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 3, 2, 1, 2, 4, 5, 3, 1, 3, 7, 12, 12, 6, 1, 3, 9, 19, 28, 25, 11, 1, 4, 14, 36, 65, 81, 63, 24, 1, 4, 16, 48, 107, 172, 193, 136, 47, 1, 5, 22, 75, 192, 369, 522, 522, 331, 103, 1, 5, 25, 96, 284, 643, 1108, 1420, 1292, 750, 214, 1, 6

Offset: 1

Gus Wiseman, Mar 19 2018

A binary enriched p-tree of weight n is either a single node of weight n, or an ordered pair of binary enriched p-trees with weakly decreasing weights summing to n.

			Triangle begins:
  1
  1   1
  1   1   1
  1   2   3   2
  1   2   4   5   3
  1   3   7  12  12   6
  1   3   9  19  28  25  11
  1   4  14  36  65  81  63  24
  1   4  16  48 107 172 193 136  47
  1   5  22  75 192 369 522 522 331 103
  ...
The T(6,3) = 7 binary enriched p-trees: ((41)1), ((32)1), (4(11)), ((31)2), ((22)2), (3(21)), ((21)3).

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Last entries of each row give A000992. Row sums are A300443.

Cf. A001190, A008284, A055277, A063834, A196545, A273873, A289501, A292050, A298422, A298426, A300354, A300439, A300442, A301344, A301364-A301367.

bintrees[n_]:=Prepend[Join@@Table[Tuples[bintrees/@ptn],{ptn,Select[IntegerPartitions[n],Length[#]===2&]}],n];
Table[Length[Select[bintrees[n],Count[#,_Integer,{-1}]===k&]],{n,13},{k,n}]

A(n)={my(v=vector(n)); for(n=1, n, v[n] = y + sum(k=1, n\2, v[k]*v[n-k])); apply(p->Vecrev(p/y), v)}
{ my(T=A(10)); for(n=1, #T, print(T[n])) } \\ Andrew Howroyd, Aug 26 2018

A320169 Number of balanced enriched p-trees of weight n.

Original entry on oeis.org

1, 2, 3, 6, 9, 20, 31, 70, 114, 243, 415, 961, 1603, 3564, 6559, 14913, 26630, 60037, 110160, 248859, 458445, 1001190, 1882350, 4220358, 7765303, 16822107, 32307240, 70081784, 133716083, 291788153, 561823990, 1230204229, 2396185727, 5176454708, 10220127290

Offset: 1

Gus Wiseman, Oct 07 2018

nonn

An enriched p-tree of weight n is either the number n itself or a finite sequence of enriched p-trees whose weights are weakly decreasing and sum to n.

A tree is balanced if all leaves have the same height.

			The a(1) = 1 through a(6) = 20 balanced enriched p-trees:
  1  2     3      4           5            6
     (11)  (21)   (22)        (32)         (33)
           (111)  (31)        (41)         (42)
                  (211)       (221)        (51)
                  (1111)      (311)        (222)
                  ((11)(11))  (2111)       (321)
                              (11111)      (411)
                              ((21)(11))   (2211)
                              ((111)(11))  (3111)
                                           (21111)
                                           (111111)
                                           ((21)(21))
                                           ((22)(11))
                                           ((31)(11))
                                           ((111)(21))
                                           ((21)(111))
                                           ((211)(11))
                                           ((111)(111))
                                           ((1111)(11))
                                           ((11)(11)(11))

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

Cf. A000311, A000669, A001678, A005804, A048816, A079500, A119262, A120803, A141268, A196545, A289501, A319312.

Cf. A316624, A320154, A320155, A320160, A320179.

eptrs[n_]:=Prepend[Join@@Table[Tuples[eptrs/@p],{p,Rest[IntegerPartitions[n]]}],n];
Table[Length[Select[eptrs[n],SameQ@@Length/@Position[#,_Integer]&]],{n,12}]

seq(n)={my(p=x/(1-x) + O(x*x^n), q=0); while(p, q+=p; p = 1/prod(k=1, n, 1 - polcoef(p,k)*x^k + O(x*x^n)) - 1 - p); Vec(q)} \\ Andrew Howroyd, Oct 26 2018

Terms a(16) and beyond from Andrew Howroyd, Oct 26 2018

A331687 Number of locally disjoint enriched p-trees of weight n.

Original entry on oeis.org

1, 2, 4, 12, 29, 93, 249, 803, 2337, 7480, 23130, 77372, 247598, 834507, 2762222

Offset: 1

Gus Wiseman, Jan 31 2020

A locally disjoint enriched p-tree of weight n is either the number n itself or a finite sequence of non-overlapping locally disjoint enriched p-trees whose weights are weakly decreasing and sum to n.

			The a(1) = 1 through a(4) = 12 enriched p-trees:
  1  2     3        4
     (11)  (21)     (22)
           (111)    (31)
           ((11)1)  (211)
                    (1111)
                    ((11)2)
                    ((21)1)
                    (2(11))
                    ((11)11)
                    ((111)1)
                    (((11)1)1)
                    ((11)(11))

The orderless version is A316696.
The identity case is A331684.
P-trees are A196545.
Enriched p-trees are A289501.
Locally disjoint identity trees are A316471.
Enriched identity p-trees are A331875.
Cf. A000669, A141268, A316473, A316495, A316694, A316697, A319312, A331678, A331679, A331680, A331686, A331871, A331874.

disjointQ[u_]:=Apply[And,Outer[#1==#2||Intersection[#1,#2]=={}&,u,u,1],{0,1}];
ldep[n_]:=Prepend[Select[Join@@Table[Tuples[ldep/@p],{p,Rest[IntegerPartitions[n]]}],disjointQ[DeleteCases[#,_Integer]]&],n];
Table[Length[ldep[n]],{n,10}]

A300866 Signed recurrence over binary strict trees: a(n) = 1 - Sum_{x + y = n, 0 < x < y < n} a(x) * a(y).

Original entry on oeis.org

1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, -1, 1, 1, -2, 3, -1, -3, 8, -8, 1, 14, -26, 22, 10, -59, 90, -52, -74, 238, -291, 80, 417, -930, 915, 124, -1991, 3483, -2533, -2148, 9011, -12596, 5754, 14350, -37975, 42735, -4046, -77924, 154374, -133903, -56529, 376844, -591197, 355941, 522978, -1706239

Offset: 0

Gus Wiseman, Mar 13 2018

sign

Andrew Howroyd, Table of n, a(n) for n = 0..500

Cf. A000992, A001190, A007317, A063834, A099323, A196545, A220418, A273866, A273873, A289501, A290261, A300442, A300443, A300862, A300863, A300864, A300865.

a[n_]:=a[n]=1-Sum[a[k]*a[n-k],{k,1,(n-1)/2}];
Array[a,40]

seq(n)={my(v=vector(n)); for(n=1, n, v[n] = 1 - sum(k=1, (n-1)\2, v[k]*v[n-k])); concat([1], v)} \\ Andrew Howroyd, Aug 27 2018

A320330 Number of T_0 multiset partitions of integer partitions of n.

Original entry on oeis.org

1, 1, 3, 5, 13, 25, 50, 100, 195, 366, 707, 1333, 2440

Offset: 0

Gus Wiseman, Oct 11 2018

The dual of a multiset partition has, for each vertex, one part consisting of the indices (or positions) of the parts containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}. The T_0 condition means the dual is strict.

			The a(1) = 1 through a(5) = 25 multiset partitions:
  {{1}}  {{2}}      {{3}}          {{4}}              {{5}}
         {{1,1}}    {{1,1,1}}      {{2,2}}            {{1,1,3}}
         {{1},{1}}  {{1},{2}}      {{1,1,2}}          {{1,2,2}}
                    {{1},{1,1}}    {{1},{3}}          {{1},{4}}
                    {{1},{1},{1}}  {{2},{2}}          {{2},{3}}
                                   {{1,1,1,1}}        {{1,1,1,2}}
                                   {{1},{1,2}}        {{1},{1,3}}
                                   {{2},{1,1}}        {{1},{2,2}}
                                   {{1},{1,1,1}}      {{2},{1,2}}
                                   {{1,1},{1,1}}      {{3},{1,1}}
                                   {{1},{1},{2}}      {{1,1,1,1,1}}
                                   {{1},{1},{1,1}}    {{1},{1,1,2}}
                                   {{1},{1},{1},{1}}  {{1,1},{1,2}}
                                                      {{1},{1},{3}}
                                                      {{1},{2},{2}}
                                                      {{2},{1,1,1}}
                                                      {{1},{1,1,1,1}}
                                                      {{1,1},{1,1,1}}
                                                      {{1},{1},{1,2}}
                                                      {{1},{2},{1,1}}
                                                      {{1},{1},{1,1,1}}
                                                      {{1},{1,1},{1,1}}
                                                      {{1},{1},{1},{2}}
                                                      {{1},{1},{1},{1,1}}
                                                      {{1},{1},{1},{1},{1}}

Cf. A001970, A047968, A050342, A089259, A141268, A261049, A289501, A305551, A316983, A319066, A319312, A320328, A320331.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
Table[Length[Select[Join@@mps/@IntegerPartitions[n],UnsameQ@@dual[#]&]],{n,8}]

A294018 Number of strict trees whose leaves are the parts of the integer partition with Heinz number n.

Original entry on oeis.org

0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 3, 1, 0, 1, 1, 1, 3, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 0, 1, 4, 1, 1, 1, 3, 1, 6, 1, 1, 1, 1, 1, 4, 1, 1, 0, 1, 1, 8, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 4, 1, 1, 4, 1, 1, 6, 1, 4, 1, 1, 1, 4, 1, 1, 1, 1, 1, 13

Offset: 1

Gus Wiseman, Feb 06 2018

nonn

By convention a(1) = 0.

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

			The a(84) = 8 strict trees: (((42)1)1), (((41)2)1), ((4(21))1), ((421)1), (((41)1)2), ((41)(21)), ((41)21), (4(21)1).

Cf. A000009, A000041, A000720, A001222, A056239, A063834, A196545, A215366, A273873, A281145, A289501, A296150, A299201, A299202, A299203.

nn=120;
ptns=Table[If[n===1,{},Join@@Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]],{n,nn}];
tris=Join@@Map[Tuples[IntegerPartitions/@#]&,ptns];
qci[y_]:=qci[y]=If[Length[y]===1,1,Sum[Times@@qci/@t,{t,Select[tris,And[Length[#]>1,Sort[Join@@#,Greater]===y,UnsameQ@@Total/@#]&]}]];
qci/@ptns

A273873(n) = Sum_{i=1..A000041(n)} a(A215366(n,i)).

cp's OEIS Frontend

A331875 Number of enriched identity p-trees of weight n.

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A290971 Write x/(1-x) in the form Sum_{j>=1} a(j)*x^j/(1+a(j)*x^j).

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A300352 Number of strict trees of weight n with distinct leaves.

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A300442 Number of binary strict trees of weight n.

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A301368 Regular triangle where T(n,k) is the number of binary enriched p-trees of weight n with k leaves.

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Mathematica

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A320169 Number of balanced enriched p-trees of weight n.

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A331687 Number of locally disjoint enriched p-trees of weight n.

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Mathematica

A300866 Signed recurrence over binary strict trees: a(n) = 1 - Sum_{x + y = n, 0 < x < y < n} a(x) * a(y).

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A290971 Write x/(1-x) in the form Sum_{j>=1} a(j)x^j/(1+a(j)x^j).