A196545 -id:A196545 - cp's OEIS frontend

A330471 Number of series/singleton-reduced rooted trees on strongly normal multisets of size n.

1, 1, 2, 9, 69, 623, 7803, 110476, 1907428

Offset: 0

Gus Wiseman, Dec 23 2019

A multiset is strongly normal if it covers an initial interval of positive integers with weakly decreasing multiplicities.

A series/singleton-reduced rooted tree on a multiset m is either the multiset m itself or a sequence of series/singleton-reduced rooted trees, one on each part of a multiset partition of m that is neither minimal (all singletons) nor maximal (only one part). This is a multiset generalization of singleton-reduced phylogenetic trees (A000311).

			The a(0) = 1 through a(3) = 9 trees:
  ()  (1)  (11)  (111)
           (12)  (112)
                 (123)
                 ((1)(11))
                 ((1)(12))
                 ((1)(23))
                 ((2)(11))
                 ((2)(13))
                 ((3)(12))
The a(4) = 69 trees, with singleton leaves (x) replaced by just x:
  (1111)      (1112)      (1122)      (1123)      (1234)
  (1(111))    (1(112))    (1(122))    (1(123))    (1(234))
  (11(11))    (11(12))    (11(22))    (11(23))    (12(34))
  ((11)(11))  (12(11))    (12(12))    (12(13))    (13(24))
  (1(1(11)))  (2(111))    (2(112))    (13(12))    (14(23))
              ((11)(12))  (22(11))    (2(113))    (2(134))
              (1(1(12)))  ((11)(22))  (23(11))    (23(14))
              (1(2(11)))  (1(1(22)))  (3(112))    (24(13))
              (2(1(11)))  ((12)(12))  ((11)(23))  (3(124))
                          (1(2(12)))  (1(1(23)))  (34(12))
                          (2(1(12)))  ((12)(13))  (4(123))
                          (2(2(11)))  (1(2(13)))  ((12)(34))
                                      (1(3(12)))  (1(2(34)))
                                      (2(1(13)))  ((13)(24))
                                      (2(3(11)))  (1(3(24)))
                                      (3(1(12)))  ((14)(23))
                                      (3(2(11)))  (1(4(23)))
                                                  (2(1(34)))
                                                  (2(3(14)))
                                                  (2(4(13)))
                                                  (3(1(24)))
                                                  (3(2(14)))
                                                  (3(4(12)))
                                                  (4(1(23)))
                                                  (4(2(13)))
                                                  (4(3(12)))

The case with all atoms different is A000311.
The case with all atoms equal is A196545.
The orderless version is A316652.
The unlabeled version is A330470.
The case where the leaves are sets is A330628.
The version for just normal (not strongly normal) is A330654.
Cf. A000669, A004114, A005121, A005804, A281118, A318812, A318848, A319312, A330465, A330467, A330475.

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]]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
mtot[m_]:=Prepend[Join@@Table[Tuples[mtot/@p],{p,Select[mps[m],Length[#]>1&&Length[#]

A330625 Number of series-reduced rooted trees whose leaves are sets (not necessarily disjoint) with multiset union a strongly normal multiset of size n.

Original entry on oeis.org

1, 1, 3, 14, 123, 1330, 19694

Offset: 0

Gus Wiseman, Dec 25 2019

A rooted tree is series-reduced if it has no unary branchings, so every non-leaf node covers at least two other nodes.

A finite multiset is strongly normal if it covers an initial interval of positive integers with weakly decreasing multiplicities.

			The a(1) = 1 through a(3) = 14 trees:
  {1}  {1,2}      {1,2,3}
       {{1},{1}}  {{1},{1,2}}
       {{1},{2}}  {{1},{2,3}}
                  {{2},{1,3}}
                  {{3},{1,2}}
                  {{1},{1},{1}}
                  {{1},{1},{2}}
                  {{1},{2},{3}}
                  {{1},{{1},{1}}}
                  {{1},{{1},{2}}}
                  {{1},{{2},{3}}}
                  {{2},{{1},{1}}}
                  {{2},{{1},{3}}}
                  {{3},{{1},{2}}}

The generalization where the leaves are multisets is A330467.
The singleton-reduced case is A330628.
The unlabeled version is A330624.
The case with all atoms distinct is A005804.
The case with all atoms equal is A196545.
The case where all leaves are singletons is A330471.
Cf. A000669, A004111, A141268, A300660, A316652, A330469, A330475, A330626, A330668, A330675.

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]]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
srtrees[m_]:=Prepend[Join@@Table[Tuples[srtrees/@p],{p,Select[mps[m],Length[#1]>1&]}],m];
Table[Sum[Length[Select[srtrees[s],FreeQ[#,{_,x_Integer,x_Integer,_}]&]],{s,strnorm[n]}],{n,0,5}]

A330628 Number of series/singleton-reduced rooted trees on strongly normal multisets of size n whose leaves are sets (not necessarily disjoint).

Original entry on oeis.org

1, 1, 1, 5, 42, 423, 5458, 80926

Offset: 0

Gus Wiseman, Dec 26 2019

A series/singleton-reduced rooted tree on a multiset m is either the multiset m itself or a sequence of series/singleton-reduced rooted trees, one on each part of a multiset partition of m that is neither minimal (all singletons) nor maximal (only one part).

A finite multiset is strongly normal if it covers an initial interval of positive integers with weakly decreasing multiplicities.

			The a(4) = 42 trees:
  {{1}{1}{12}}    {{12}{12}}      {{1}{123}}      {1234}
  {{1}{{1}{12}}}  {{1}{2}{12}}    {{12}{13}}      {{1}{234}}
                  {{1}{{2}{12}}}  {{1}{1}{23}}    {{12}{34}}
                  {{2}{{1}{12}}}  {{1}{2}{13}}    {{13}{24}}
                                  {{1}{3}{12}}    {{14}{23}}
                                  {{1}{{1}{23}}}  {{2}{134}}
                                  {{1}{{2}{13}}}  {{3}{124}}
                                  {{1}{{3}{12}}}  {{4}{123}}
                                  {{2}{{1}{13}}}  {{1}{2}{34}}
                                  {{3}{{1}{12}}}  {{1}{3}{24}}
                                                  {{1}{4}{23}}
                                                  {{2}{3}{14}}
                                                  {{2}{4}{13}}
                                                  {{3}{4}{12}}
                                                  {{1}{{2}{34}}}
                                                  {{1}{{3}{24}}}
                                                  {{1}{{4}{23}}}
                                                  {{2}{{1}{34}}}
                                                  {{2}{{3}{14}}}
                                                  {{2}{{4}{13}}}
                                                  {{3}{{1}{24}}}
                                                  {{3}{{2}{14}}}
                                                  {{3}{{4}{12}}}
                                                  {{4}{{1}{23}}}
                                                  {{4}{{2}{13}}}
                                                  {{4}{{3}{12}}}

The generalization where leaves are multisets is A330471.
The non-singleton-reduced version is A330625.
The unlabeled version is A330626.
The case with all atoms distinct is A000311.
Strongly normal multiset partitions are A035310.
Cf. A000669, A004111, A004114, A005804, A196545, A281118, A330465, A330467, A330624, A330654, A330668.

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]]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
ssrtrees[m_]:=Prepend[Join@@Table[Tuples[ssrtrees/@p],{p,Select[mps[m],Length[m]>Length[#1]>1&]}],m];
Table[Sum[Length[Select[ssrtrees[s],FreeQ[#,{_,x_Integer,x_Integer,_}]&]],{s,strnorm[n]}],{n,0,5}]

A292137 G.f.: Im(1/(i*x; x)_inf), where (a; q)_inf is the q-Pochhammer symbol, i = sqrt(-1).

Original entry on oeis.org

0, 1, 1, 0, 0, 0, -1, -2, -2, -2, -2, -3, -3, -2, -2, -2, -1, 1, 2, 2, 4, 6, 7, 8, 10, 13, 14, 14, 15, 17, 17, 15, 15, 16, 14, 10, 8, 6, 1, -5, -10, -14, -21, -31, -38, -43, -53, -64, -71, -77, -86, -97, -104, -108, -115, -124, -127, -125, -127, -130, -125, -116

Offset: 0

Seiichi Manyama, Sep 09 2017

			Product_{k>=1} 1/(1 - i*x^k) = 1 + (0+1i)*x + (-1+1i)*x^2 + (-1+0i)*x^3 + (-1+0i)*x^4 + (-1+0i)*x^5 + (-2-1i)*x^6 + (-1-2i)*x^7 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, q-Pochhammer Symbol.

Cf. A000108, A010815, A027193, A063834, A067659, A081362, A099323, A196545, A220418, A290261, A292042, A292043, A292136, A292138, A298118, A300355.

N:= 100:
S := convert(series( add( (-1)^n*x^(2*n+1)/(mul(1 - x^k,k = 1..2*n+1)), n = 0..N ), x, N+1 ), polynom):
seq(coeff(S, x, n), n = 0..N); # Peter Bala, Jan 15 2021

Im[CoefficientList[Series[1/QPochhammer[I*x, x], {x, 0, 100}], x]] (* Vaclav Kotesovec, Sep 17 2017 *)

1/(i*x; x)_inf is the g.f. for A292136(n) + i*a(n).
a(n) = Sum (-1)^((k - 1)/2) where the sum is over all integer partitions of n into an odd number of parts and k is the number of parts. - Gus Wiseman, Mar 08 2018
G.f.: Sum_{n >= 0} (-1)^n * x^(2*n+1)/Product_{k = 1..2*n+1} (1 - x^k). - Peter Bala, Jan 15 2021

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)).

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

Original entry on oeis.org

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)).

A318848 Number of complete tree-partitions of a multiset whose multiplicities are the prime indices of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 5, 4, 12, 9, 12, 17, 34, 29, 44, 26, 92, 90, 277, 68, 171, 93, 806, 144, 197, 309, 581, 269, 2500, 428, 7578, 236, 631, 1025, 869, 954, 24198, 3463, 2402, 712, 75370, 1957, 243800, 1040, 3200, 11705, 776494, 1612, 4349, 2358, 8862, 3993, 2545777

Offset: 1

Gus Wiseman, Sep 04 2018

nonn

This multiset is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

A tree-partition of m is either m itself or a sequence of tree-partitions, one of each part of a multiset partition of m with at least two parts. A tree-partition is complete if the leaves are all multisets of length 1.

			The a(12) = 17 complete tree-partitions of {1,1,2,3} with the leaves (x) replaced with just x:
  (1(1(23)))
  (1(2(13)))
  (1(3(12)))
  (2(1(13)))
  (2(3(11)))
  (3(1(12)))
  (3(2(11)))
  ((11)(23))
  ((12)(13))
  (1(123))
  (2(113))
  (3(112))
  (11(23))
  (12(13))
  (13(12))
  (23(11))
  (1123)

Cf. A000311, A001055, A196545, A281118, A281119, A305936, A318762, A318812, A318813, A318846, A318847, A318849.

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]]]];
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
allmsptrees[m_]:=Prepend[Join@@Table[Tuples[allmsptrees/@p],{p,Select[mps[m],Length[#]>1&]}],m];
Table[Length[Select[allmsptrees[nrmptn[n]],FreeQ[#,{?AtomQ,_}]&]],{n,20}]

a(n) = A281119(A181821(n)).
a(prime(n)) = A196545(n)
a(2^n) = A000311(n).

More terms from Jinyuan Wang, Jun 26 2020

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.

cp's OEIS Frontend

A330471 Number of series/singleton-reduced rooted trees on strongly normal multisets of size n.

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A330625 Number of series-reduced rooted trees whose leaves are sets (not necessarily disjoint) with multiset union a strongly normal multiset of size n.

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A330628 Number of series/singleton-reduced rooted trees on strongly normal multisets of size n whose leaves are sets (not necessarily disjoint).

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A292137 G.f.: Im(1/(i*x; x)_inf), where (a; q)_inf is the q-Pochhammer symbol, i = sqrt(-1).

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

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

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

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Mathematica

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

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