A196545 -id:A196545 - cp's OEIS frontend

A280994 Triangle read by rows giving Matula-Goebel numbers of planted achiral trees with n nodes.

1, 2, 3, 4, 5, 7, 8, 9, 11, 16, 17, 19, 23, 31, 32, 53, 59, 67, 25, 27, 49, 64, 83, 127, 131, 241, 277, 331, 97, 103, 128, 227, 311, 431, 709, 739, 1523, 1787, 2221, 81, 121, 256, 289, 361, 509, 563, 719, 1433, 2063, 3001, 5381, 5623, 12763, 15299, 19577

Offset: 1

Gus Wiseman, Jan 12 2017

An achiral tree is either (case 1) a single node or (case 2) a finite constant sequence (t,t,..,t) of achiral trees. Only in case 2 is an achiral tree considered to be a generalized Bethe tree (according to A214577).

			Triangle begins:
1,
2,
3, 4,
5, 7, 8,
9, 11, 16, 17, 19,
23, 31, 32, 53, 59, 67,
25, 27, 49, 64, 83, 127, 131, 241, 277, 331.

E. Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288 [math.CO], 2011.

Cf. A003238 (row lengths), A214577, A061773, A061775, A004111, A007097, A196545, A275870.

nn=7;MGNumber[[]]:=1;MGNumber[x:[__]]:=If[Length[x]===1,Prime[MGNumber[x[[1]]]],Times@@Prime/@MGNumber/@x];
cits[n_]:=If[n===1,{1},Join@@Table[ConstantArray[#,(n-1)/d]&/@cits[d],{d,Divisors[n-1]}]];
Table[Sort[MGNumber/@(cits[n]/.(1->{}))],{n,nn}]

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.

A300440 Number of odd strict trees of weight n (all outdegrees are odd).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 3, 5, 7, 11, 18, 27, 45, 75, 125, 207, 353, 591, 1013, 1731, 2984, 5122, 8905, 15369, 26839, 46732, 81850, 142932, 251693, 441062, 778730, 1370591, 2425823, 4281620, 7601359, 13447298, 23919512, 42444497, 75632126, 134454505, 240100289

Offset: 1

Gus Wiseman, Mar 05 2018

nonn

An odd strict tree of weight n is either a single node of weight n, or a finite odd-length sequence of at least 3 odd strict trees with strictly decreasing weights summing to n.

			The a(10) = 7 odd strict trees: 10, (721), (631), (541), (532), ((421)21), ((321)31).

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

Cf. A000009, A000992, A032305, A063834, A078408, A089259, A196545, A273873, A279785, A289501, A298118, A300301, A300352, A300353, A300436, A300439.

g[n_]:=g[n]=1+Sum[Times@@g/@y,{y,Select[IntegerPartitions[n],Length[#]>1&&OddQ[Length[#]]&&UnsameQ@@#&]}];
Array[g,20]

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

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

A306187 Number of n-times partitions of n.

Original entry on oeis.org

1, 1, 3, 10, 65, 371, 3780, 33552, 472971, 5736082, 97047819, 1547576394, 32992294296, 626527881617, 15202246707840, 352290010708120, 9970739854456849, 262225912049078193, 8309425491887714632, 250946978120046026219, 8898019305511325083149

Offset: 0

Alois P. Heinz, Jan 27 2019

nonn

A k-times partition of n for k > 1 is a sequence of (k-1)-times partitions, one of each part in an integer partition of n. A 1-times partition of n is just an integer partition of n. The only 0-times partition of n is the number n itself. - Gus Wiseman, Jan 27 2019

			From _Gus Wiseman_, Jan 27 2019: (Start)
The a(1) = 1 through a(3) = 10 partitions:
  (1)  ((2))     (((3)))
       ((11))    (((21)))
       ((1)(1))  (((111)))
                 (((2)(1)))
                 (((11)(1)))
                 (((2))((1)))
                 (((1)(1)(1)))
                 (((11))((1)))
                 (((1)(1))((1)))
                 (((1))((1))((1)))
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..410
Wikipedia, Partition (number theory)

Main diagonal of A323718.
Cf. A000041, A306188.
Cf. A001970, A063834, A096752, A196545, A261280, A289501, A290354, A327619.

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$3):
seq(a(n), n=0..25);

ptnlevct[n_,k_]:=Switch[k,0,1,1,PartitionsP[n],_,SeriesCoefficient[Product[1/(1-ptnlevct[m,k-1]*x^m),{m,n}],{x,0,n}]];
Table[ptnlevct[n,n],{n,0,8}] (* Gus Wiseman, Jan 27 2019 *)

a(n) = A323718(n,n).

A330654 Number of series/singleton-reduced rooted trees on normal multisets of size n.

Original entry on oeis.org

1, 1, 2, 12, 112, 1444, 24099, 492434, 11913985

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 normal if it covers an initial interval of positive integers.
First differs from A316651 at a(6) = 24099, A316651(6) = 24086. For example, ((1(12))(2(11))) and ((2(11))(1(12))) are considered identical for A316651 (series-reduced rooted trees), but {{{1},{1,2}},{{2},{1,1}}} and {{{2},{1,1}},{{1},{1,2}}} are different series/singleton-reduced rooted trees.

			The a(0) = 1 through a(3) = 12 trees:
  {}  {1}  {1,1}  {1,1,1}
           {1,2}  {1,1,2}
                  {1,2,2}
                  {1,2,3}
                  {{1},{1,1}}
                  {{1},{1,2}}
                  {{1},{2,2}}
                  {{1},{2,3}}
                  {{2},{1,1}}
                  {{2},{1,2}}
                  {{2},{1,3}}
                  {{3},{1,2}}

The orderless version is A316651.
The strongly normal case is A330471.
The unlabeled version is A330470.
The balanced version is A330655.
The case with all atoms distinct is A000311.
The case with all atoms equal is A196545.
Normal multiset partitions are A255906.
Cf. A000669, A004114, A005804, A281118, A316651, A330469, A330626, A330676.

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]]]];
allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
ssrtrees[m_]:=Prepend[Join@@Table[Tuples[ssrtrees/@p],{p,Select[mps[m],Length[m]>Length[#1]>1&]}],m];
Table[Sum[Length[ssrtrees[s]],{s,allnorm[n]}],{n,0,5}]

A331684 Number of locally disjoint enriched identity p-trees of weight n.

Original entry on oeis.org

1, 1, 2, 3, 6, 14, 30, 68, 157, 379, 901, 2229, 5488, 13846, 34801, 89368, 228186, 592943, 1533511, 4026833

Offset: 1

Gus Wiseman, Jan 31 2020

A locally disjoint enriched identity p-tree of weight n is either the number n itself or a finite sequence of distinct non-overlapping locally disjoint 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)

The orderless version is A316694.
The non-identity version is A331687.
Identity trees are A004111.
P-trees are A196545.
Enriched p-trees are A289501.
Locally disjoint identity trees are A316471.
Enriched identity p-trees are A331875, with locally disjoint case A331687.
Cf. A000669, A005804, A141268, A300660, A316696, A316697, A331678, A331679, A331680, A331683, A331686, A331783, A331874.

disjointQ[u_]:=Apply[And,Outer[#1==#2||Intersection[#1,#2]=={}&,u,u,1],{0,1}];
ldeip[n_]:=Prepend[Select[Join@@Table[Tuples[ldeip/@p],{p,Rest[IntegerPartitions[n]]}],UnsameQ@@#&&disjointQ[DeleteCases[#,_Integer]]&],n];
Table[Length[ldeip[n]],{n,12}]

A294080 Same-tree Moebius function of the multiorder of integer partitions indexed by Heinz numbers.

Original entry on oeis.org

0, 1, 1, -1, 1, 0, 1, -1, -1, 0, 1, 2, 1, 0, 0, -2, 1, 0, 1, 0, 0, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, -1, 0, 0, 0, 3, 1, 0, 0, 2, 1, 0, 1, 0, 0, 0, 1, -3, -1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 2, -1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, -2, 0, 1, -4, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 8

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).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000041, A056239, A063834, A196545, A273873, A281145, A289501, A294018, A294019, A296150, 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];
rmu[y_]:=rmu[y]=If[Length[y]===1,1,-Sum[Times@@rmu/@t,{t,Select[tris,And[Length[#]>1,Sort[Join@@#,Greater]===y,SameQ@@Total/@#]&]}]];
rmu/@ptns

A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i,2] * primepi(f[i,1]))); }
muifbalancedfactorization(v) = if(!#v, 1, my(pw=A056239(v[1]), m=-1); for(i=1,#v,if(A056239(v[i])!=pw,return(0), m *= A294080(v[i]))); (m));
A294080aux(n, m, facs) = if(1==n, muifbalancedfactorization(Vec(facs)), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); s += A294080aux(n/d, m, newfacs))); (s));
A294080(n) = if(1==n,0,if(isprime(n),1,A294080aux(n, n-1, List([]))));
\\ A memoized implementation:
map294080 = Map();
A294080(n) = if(1==n,0,if(isprime(n),1,if(mapisdefined(map294080,n), mapget(map294080,n), my(v=A294080aux(n, n-1, List([]))); mapput(map294080,n,v); (v)))); \\ Antti Karttunen, Sep 22 2018

mu(y) = Sum_{g(t)=y} (-1)^d(t), where the sum is over all same-trees (A281145, A294019) whose multiset of leaves is the integer partition y, and d(t) is the number of non-leaf nodes in t.

A300864 Signed recurrence over strict trees: a(n) = -1 + Sum_{y1 + ... + yk = n, y1 > ... > yk > 0, k > 1} a(y1) * ... * a(yk).

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 2, -1, 0, 4, -6, 6, 6, -24, 38, -17, -64, 188, -230, -6, 662, -1432, 1286, 1210, -6362, 10692, -5530, -18274, 57022, -74364, 174, 216703, -489544, 467860, 391258, -2256430, 3948206, -2234064, -6725362, 21920402, -29716570, 2095564, 84595798, -198418242, 197499846

Offset: 1

Gus Wiseman, Mar 13 2018

sign

Cf. A000992, A018819, A063834, A099323, A196545, A220418, A273866, A273873, A289501, A290261, A300862, A300863, A300865, A300866.

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

A300865 Signed recurrence over binary enriched p-trees: a(n) = (-1)^(n-1) + Sum_{x + y = n, 0 < x <= y < n} a(x) * a(y).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 2, 2, 4, 6, 10, 16, 27, 46, 77, 131, 224, 391, 672, 1180, 2050, 3626, 6344, 11276, 19863, 35479, 62828, 112685, 200462, 360627, 644199, 1162296, 2083572, 3768866, 6777314, 12289160, 22158106, 40255496, 72765144, 132453122, 239936528, 437445448

Offset: 1

Gus Wiseman, Mar 13 2018

nonn

Cf. A000992, A001190, A007317, A063834, A099323, A196545, A220418, A273866, A273873, A289501, A290261, A300352, A300442, A300443, A300862, A300863, A300864, A300866.

a[n_]:=a[n]=(-1)^(n-1)+Sum[a[k]*a[n-k],{k,1,n/2}];
Array[a,50]

cp's OEIS Frontend

A280994 Triangle read by rows giving Matula-Goebel numbers of planted achiral trees with n nodes.

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

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A300440 Number of odd strict trees of weight n (all outdegrees are odd).

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PARI

A301595 Number of thrice-partitions of n.

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Maple

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A306187 Number of n-times partitions of n.

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Maple

Mathematica

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A330654 Number of series/singleton-reduced rooted trees on normal multisets of size n.

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Mathematica

A331684 Number of locally disjoint enriched identity p-trees of weight n.

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Mathematica

A294080 Same-tree Moebius function of the multiorder of integer partitions indexed by Heinz numbers.

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