A289501 -id:A289501 - cp's OEIS frontend

A300353 Number of strict trees of weight n with odd leaves.

1, 1, 0, 1, 1, 2, 2, 4, 7, 14, 24, 46, 92, 186, 368, 750, 1529, 3160, 6510, 13590, 28374, 59780, 125732, 266468, 564188, 1202842, 2560106, 5484304, 11732400, 25229068, 54187918, 116938702, 252039411, 545593378, 1179545874, 2560009400, 5550315640, 12075064432

Offset: 0

Gus Wiseman, Mar 03 2018

nonn

This sequence is initially dominated by A300352 but eventually becomes much greater.

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) = 7 strict trees with odd leaves: (71), (53), (((51)1)1), (((31)3)1), (((31)1)3), ((31)31), (((((31)1)1)1)1).

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

Cf. A000009, A063834, A078408, A089259, A196545, A279374, A279785, A289501, A294018, A294079, A299203, A300300, A300301, A300351, A300352, A300354, A300355.

d[n_]:=d[n]=If[EvenQ[n],0,1]+Sum[Times@@d/@y,{y,Select[IntegerPartitions[n],Length[#]>1&&UnsameQ@@#&]}];
Table[d[n],{n,40}]

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

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

a(n) = Sum_{i=1..A000009(n)} A294018(A300351(n,i)).

A300354 Number of enriched p-trees of weight n with distinct leaves.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 8, 8, 13, 17, 54, 56, 98, 125, 195, 500, 606, 921, 1317, 1912, 2635, 6667, 7704, 12142, 16958, 24891, 33388, 47792, 106494, 126475, 195475, 268736, 393179, 523775, 750251, 979518, 2090669, 2457315, 3759380, 5066524, 7420874, 9726501, 13935546

Offset: 0

Gus Wiseman, Mar 03 2018

nonn

An enriched p-tree of weight n > 0 is either a single node of weight n, or a sequence of two or more enriched p-trees with weakly decreasing weights summing to n.

			The a(6) = 8 enriched p-trees with distinct leaves: 6, (42), (51), ((31)2), ((32)1), (3(21)), ((21)3), (321).

Cf. A000009, A000041, A063834, A196545, A246867, A273873, A281145, A289501, A290261, A294018, A296150, A299201, A299202, A299203, A300352, A300353, A300355.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
ept[q_]:=ept[q]=If[Length[q]===1,1,Total[Times@@@Map[ept,Join@@Function[sptn,Join@@@Tuples[Permutations/@GatherBy[sptn,Total]]]/@Select[sps[q],Length[#]>1&],{2}]]];
Table[Total[ept/@Select[IntegerPartitions[n],UnsameQ@@#&]],{n,1,30}]

a(n) = Sum_{i=1..A000009(n)} A299203(A246867(n,i)).

A300355 Number of enriched p-trees of weight n with odd leaves.

Original entry on oeis.org

1, 1, 1, 3, 6, 16, 47, 132, 410, 1254, 4052, 12818, 42783, 139082, 469924, 1563606, 5353966, 18065348, 62491018, 213391790, 743836996, 2565135934, 8994087070, 31251762932, 110245063771, 385443583008, 1365151504722, 4800376128986, 17070221456536, 60289267885410

Offset: 0

Gus Wiseman, Mar 03 2018

nonn

An enriched p-tree of weight n > 0 is either a single node of weight n, or a sequence of two or more enriched p-trees with weakly decreasing weights summing to n.

			The a(5) = 16 enriched p-trees of weight with odd leaves:
5,
((31)1), ((((11)1)1)1), (((111)1)1), (((11)(11))1), (((11)11)1), ((1111)1),
(3(11)), (((11)1)(11)), ((111)(11)),
(311), (((11)1)11), ((111)11),
((11)(11)1),
((11)111),
(11111).

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

Cf. A000009, A063834, A078408, A089259, A196545, A279374, A279785, A289501, A294079, A299202, A299203, A300300, A300301, A300352, A300353, A300354.

c[n_]:=c[n]=If[EvenQ[n],0,1]+Sum[Times@@c/@y,{y,Select[IntegerPartitions[n],Length[#]>1&]}];
Table[c[n],{n,30}]

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

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

a(n) = Sum_{i=1..A000009(n)} A299203(A300351(n,i)).

A320331 Number of strict T_0 multiset partitions of integer partitions of n.

Original entry on oeis.org

1, 1, 2, 4, 8, 17, 30, 61, 110, 207, 381, 711, 1250

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) = 17 multiset partitions:
  {{1}}  {{2}}    {{3}}        {{4}}          {{5}}
         {{1,1}}  {{1,1,1}}    {{2,2}}        {{1,1,3}}
                  {{1},{2}}    {{1,1,2}}      {{1,2,2}}
                  {{1},{1,1}}  {{1},{3}}      {{1},{4}}
                               {{1,1,1,1}}    {{2},{3}}
                               {{1},{1,2}}    {{1,1,1,2}}
                               {{2},{1,1}}    {{1},{1,3}}
                               {{1},{1,1,1}}  {{1},{2,2}}
                                              {{2},{1,2}}
                                              {{3},{1,1}}
                                              {{1,1,1,1,1}}
                                              {{1},{1,1,2}}
                                              {{1,1},{1,2}}
                                              {{2},{1,1,1}}
                                              {{1},{1,1,1,1}}
                                              {{1,1},{1,1,1}}
                                              {{1},{2},{1,1}}

Cf. A001970, A047968, A050342, A089259, A141268, A261049, A289501, A305551, A319066, A319312, A320328, A320330.

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],And[UnsameQ@@#,UnsameQ@@dual[#]]&]],{n,8}]

A290262 Irregular triangle read by rows: rows give the (negated) nonzero coefficients of t in each term of the inverse power product expansion of 1 - t * x/(1-x).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 3, 4, 2, 1, 3, 5, 5, 3, 1, 1, 4, 9, 13, 13, 9, 4, 1, 1, 4, 9, 13, 13, 9, 4, 1, 1, 5, 14, 25, 30, 24, 12, 3, 1, 5, 15, 30, 42, 42, 30, 15, 5, 1, 1, 6, 21, 48, 75, 81, 60, 30, 10, 2

Offset: 1

Gus Wiseman, Jul 24 2017

Row sums are A290261(n). A regular version is A290320.

			Triangle begins:
  1;
  1,  1;
  1,  1,
  1,  2,  2,  1;
  1,  2,  2,  1;
  1,  3,  4,  2;
  1,  3,  5,  5,  3,  1;
  1,  4,  9, 13, 13,  9,  4,  1;
  1,  4,  9, 13, 13,  9,  4,  1;
  1,  5, 14, 25, 30, 24, 12,  3;
  1,  5, 15, 30, 42, 42, 30, 15,  5,  1;
  1,  6, 21, 48, 75, 81, 60, 30, 10,  2;

Cf. A220418, A273866, A289501, A290261, A290320.

eptrees[n_]:=Prepend[Join@@Table[Tuples[eptrees/@y],{y,Rest[IntegerPartitions[n]]}],n];
eptrans[a_][n_]:=Sum[(-1)^Count[tree,_List,{0,Infinity}]*Product[a[i],{i,Flatten[{tree}]}],{tree,eptrees[n]}];
Table[DeleteCases[CoefficientList[-eptrans[-t&][n],t],0],{n,12}]

A300647 Number of same-trees of weight n in which all outdegrees are odd.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 2, 1, 10, 2, 2, 2, 2, 2, 42, 1, 2, 10, 2, 2, 138, 2, 2, 2, 34, 2, 1514, 2, 2, 42, 2, 1, 2058, 2, 162, 10, 2, 2, 8202, 2, 2, 138, 2, 2, 207370, 2, 2, 2, 130, 34, 131082, 2, 2, 1514, 2082, 2, 524298, 2, 2, 42, 2, 2, 14725738, 1, 8226, 2058, 2

Offset: 1

Gus Wiseman, Mar 10 2018

nonn

A same-tree of weight n > 0 is either a single node of weight n, or a finite sequence of two or more same-trees whose weights are all equal and sum to n.

			The a(9) = 10 odd same-trees:
9,
(333),
(33(111)), (3(111)3), ((111)33)
(3(111)(111)), ((111)3(111)), ((111)(111)3),
((111)(111)(111)), (111111111).

Antti Karttunen, Table of n, a(n) for n = 1..8192

Cf. A003238, A006241, A063834, A069283, A273873, A281145, A289078, A289079, A289501, A298118, A300436, A300439, A300574, A300648, A300649, A300650.

a[n_]:=1+Sum[a[n/d]^d,{d,Select[Rest[Divisors[n]],OddQ]}];
Array[a,80]

a(n) = if (n==1, 1, 1+sumdiv(n, d, if ((d > 1) && (d % 2), a(n/d)^d))); \\ Michel Marcus, Mar 10 2018

a(n) = 1 + Sum_d a(n/d)^d where the sum is over odd divisors of n greater than 1.

A300863 Signed recurrence over enriched p-trees: a(n) = (-1)^(n - 1) + Sum_{y1 + ... + yk = n, y1 >= ... >= yk > 0, k > 1} a(y1) * ... * a(yk).

Original entry on oeis.org

1, 0, 2, 2, 6, 14, 34, 82, 214, 566, 1482, 4058, 10950, 30406, 83786, 235714, 658286, 1874254, 5293674, 15189810, 43312542, 125075238, 359185586, 1043712922, 3015569582, 8800146182, 25565402802, 74918274562, 218572345718, 642783954238, 1882606578002

Offset: 1

Gus Wiseman, Mar 13 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..500

Cf. A000992, A063834, A099323, A196545, A220418, A273866, A273873, A289501, A290261, A300862, A300864, A300865, A300866.

a[n_]:=a[n]=(-1)^(n-1)+Sum[Times@@a/@y,{y,Select[IntegerPartitions[n],Length[#]>1&]}];
Array[a,40]

O.g.f.: (-1/(1+x) + Product 1/(1-a(n)x^n))/2.

A290973 Write 2*x/(1-x) in the form Sum_{j>=1} ((1-x^j)^a(j) - 1).

Original entry on oeis.org

-2, 1, 2, 3, 4, 6, 6, 10, 8, 15, 10, 25, 12, 28, 10, 60, 16, 25, 18, 125, 0, 66, 22, 218, 24, 91, -30, 420, 28, -387, 30, 2011, -88, 153, 28, -1894, 36, 190, -182, 8902, 40, -3234, 42, 2398, -132, 276, 46, 2340, 48, -2678, -510, 4641, 52, -1754, -198, 108400

Offset: 1

Gus Wiseman, Aug 16 2017

sign

			2x/(1-x) = (1-x)^(-2) - 1 + (1-x^2)^1 - 1 + (1-x^3)^2 - 1 + (1-x^4)^3 - 1 + ...

Cf. A048272, A220418, A260685, A281145, A289078, A289501, A290261, A290971.

a:= n-> add(binomial(n/d-1-a(d), n/d), d=
        numtheory[divisors](n) minus {n})-2:
seq(a(n), n=1..60);  # Alois P. Heinz, Aug 27 2017

nn=60;
rus=SolveAlways[Normal[Series[2x/(1-x)==Sum[(1-x^n)^a[n]-1,{n,nn}],{x,0,nn}]],x];
Array[a,nn]/.First[rus]

For all n > 0 we have: 2 = Sum_{d|n} binomial(-a(d) + n/d - 1, n/d).

A294079 Strict Moebius function of the multiorder of integer partitions indexed by Heinz numbers.

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, 1, 1, 0, -1, -1, -1, -1, 1, -1, -1, -1, 1, 2, 1, 1, 1, -1, 1, 1, 0, 1, -1, 1, 1, -1, -1, -1, -1, -1, 1, -3, 1, -1, 1, 0, -1, 2, 1, 1, -1, 1, 1, 2, 1, -1, 1, 1, -1, 2, 1, 1, 0, -1, 1, -2, -1, -1, -1, -1, 1, -3, -1

Offset: 1

Gus Wiseman, Feb 07 2018

sign

By convention a(1) = 0.

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

Cf. A000041, A000720, A056239, A063834, A196545, A273873, A289501, A294018, A294019, 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];
qmu[y_]:=qmu[y]=If[Length[y]===1,1,-Sum[Times@@qmu/@t,{t,Select[tris,And[Length[#]>1,Sort[Join@@#,Greater]===y,UnsameQ@@#]&]}]];
qmu/@ptns

mu(y) = Sum_{g(t)=y} (-1)^d(t), where the sum is over all strict trees (A273873) whose multiset of leaves is the integer partition y, and d(t) is the number of non-leaf nodes in t.

A295632 Write 1/Product_{n > 1}(1 - 1/n^s) in the form Product_{n > 1}(1 + a(n)/n^s).

Original entry on oeis.org

1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 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, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1

Offset: 2

Gus Wiseman, Nov 24 2017

sign

First negative entry is a(1024) = -4.

Cf. A001055, A045778, A050376, A220418, A220420, A273866, A273873, A289501, A290261, A290262, A290971, A290973, A295279, A295635, A295636.

nn=100;
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Solve[Table[Length[facs[n]]==Sum[Times@@a/@f,{f,Select[facs[n],UnsameQ@@#&]}],{n,2,nn}],Table[a[n],{n,2,nn}]][[1,All,2]]

cp's OEIS Frontend

A300353 Number of strict trees of weight n with odd leaves.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A300354 Number of enriched p-trees of weight n with distinct leaves.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A300355 Number of enriched p-trees of weight n with odd leaves.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A320331 Number of strict T_0 multiset partitions of integer partitions of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A290262 Irregular triangle read by rows: rows give the (negated) nonzero coefficients of t in each term of the inverse power product expansion of 1 - t * x/(1-x).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A300647 Number of same-trees of weight n in which all outdegrees are odd.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A300863 Signed recurrence over enriched p-trees: a(n) = (-1)^(n - 1) + Sum_{y1 + ... + yk = n, y1 >= ... >= yk > 0, k > 1} a(y1) * ... * a(yk).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A290973 Write 2*x/(1-x) in the form Sum_{j>=1} ((1-x^j)^a(j) - 1).

Views

Author

Keywords

Examples

Crossrefs