A005804 -id:A005804 - cp's OEIS frontend

A320175 Number of series-reduced rooted trees whose leaves are strict integer partitions whose multiset union is an integer partition of n.

1, 2, 5, 13, 37, 120, 395, 1381, 4931, 18074, 67287, 254387, 972559, 3756315, 14629237, 57395490, 226613217, 899773355, 3590349661, 14390323014, 57907783039, 233867667197, 947601928915, 3851054528838, 15693587686823, 64114744713845, 262543966114921, 1077406218930902

Offset: 1

Gus Wiseman, Oct 07 2018

nonn

A rooted tree is series-reduced if every non-leaf node has at least two branches.

			The a(1) = 1 through a(4) = 13 trees:
  (1)  (2)       (3)            (4)
       ((1)(1))  (21)           (31)
                 ((1)(2))       ((1)(3))
                 ((1)(1)(1))    ((2)(2))
                 ((1)((1)(1)))  ((1)(21))
                                ((1)(1)(2))
                                ((1)((1)(2)))
                                ((2)((1)(1)))
                                ((1)(1)(1)(1))
                                ((1)((1)(1)(1)))
                                ((1)(1)((1)(1)))
                                (((1)(1))((1)(1)))
                                ((1)((1)((1)(1))))

Andrew Howroyd, Table of n, a(n) for n = 1..200

Cf. A000081, A000311, A000669, A001678, A005804, A141268, A292504, A300660, A317099, A319312, A320173, A320174.

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]]]];
sot[m_]:=If[UnsameQ@@m,Prepend[#,m],#]&[Join@@Table[Union[Sort/@Tuples[sot/@p]],{p,Select[mps[m],Length[#]>1&]}]];
Table[Length[Join@@Table[sot[m],{m,IntegerPartitions[n]}]],{n,10}]

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(p=prod(k=1, n, 1 + x^k + O(x*x^n)), v=vector(n)); for(n=1, n, v[n]=polcoef(p, n) + EulerT(v[1..n])[n]); v} \\ Andrew Howroyd, Oct 25 2018

Terms a(11) and beyond from Andrew Howroyd, Oct 25 2018

A330668 Number of non-isomorphic balanced reduced multisystems of weight n whose leaves (which are multisets of atoms) are all sets.

Original entry on oeis.org

1, 1, 1, 3, 22, 204, 2953

Offset: 0

Gus Wiseman, Dec 27 2019

A balanced reduced multisystem is either a finite multiset, or a multiset partition with at least two parts, not all of which are singletons, of a balanced reduced multisystem. The weight of an atom is 1, while the weight of a multiset is the sum of weights of its elements.

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 22 multisystems:
  {1}  {1,2}  {1,2,3}      {1,2,3,4}
              {{1},{1,2}}  {{1},{1,2,3}}
              {{1},{2,3}}  {{1,2},{1,2}}
                           {{1,2},{1,3}}
                           {{1},{2,3,4}}
                           {{1,2},{3,4}}
                           {{1},{1},{1,2}}
                           {{1},{1},{2,3}}
                           {{1},{2},{1,2}}
                           {{1},{2},{1,3}}
                           {{1},{2},{3,4}}
                           {{{1}},{{1},{1,2}}}
                           {{{1}},{{1},{2,3}}}
                           {{{1,2}},{{1},{1}}}
                           {{{1}},{{2},{1,2}}}
                           {{{1,2}},{{1},{2}}}
                           {{{1}},{{2},{1,3}}}
                           {{{1,2}},{{1},{3}}}
                           {{{1}},{{2},{3,4}}}
                           {{{1,2}},{{3},{4}}}
                           {{{2}},{{1},{1,3}}}
                           {{{2,3}},{{1},{1}}}

The case with all atoms different is A318813.
The version where the leaves are multisets is A330474.
The tree version is A330626.
The maximum-depth case is A330677.
Unlabeled series-reduced rooted trees whose leaves are sets are A330624.
Cf. A000311, A004114, A005121, A005804, A007716, A048816, A141268, A283877, A306186, A318812, A320154, A330470, A330628, A330663.

A320171 Number of series-reduced rooted identity trees whose leaves are integer partitions whose multiset union is an integer partition of n.

Original entry on oeis.org

1, 2, 5, 11, 29, 82, 247, 782, 2579, 8702, 29975, 104818, 371111, 1327307, 4788687, 17404838, 63669763, 234237605, 866090021, 3216738344, 11995470691, 44894977263, 168582174353, 634939697164, 2398004674911, 9079614633247, 34458722286825, 131059771522401

Offset: 1

Gus Wiseman, Oct 07 2018

nonn

A rooted tree is series-reduced if every non-leaf node has at least two branches.

In an identity tree, all branches directly under any given node are different.

			The a(1) = 1 through a(4) = 11 rooted identity trees:
  (1)  (2)   (3)        (4)
       (11)  (21)       (22)
             (111)      (31)
             ((1)(2))   (211)
             ((1)(11))  (1111)
                        ((1)(3))
                        ((1)(21))
                        ((2)(11))
                        ((1)(111))
                        ((1)((1)(2)))
                        ((1)((1)(11)))

Andrew Howroyd, Table of n, a(n) for n = 1..200

Cf. A000081, A000311, A000669, A001678, A005804, A141268, A292504, A300660, A319312, A320172, A320177, A320178.

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]]]];
gig[m_]:=Prepend[Join@@Table[Union[Sort/@Select[Sort/@Tuples[gig/@mtn],UnsameQ@@#&]],{mtn,Select[mps[m],Length[#]>1&]}],m];
Table[Sum[Length[gig[y]],{y,IntegerPartitions[n]}],{n,8}]

WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}
seq(n)={my(v=vector(n)); for(n=1, n, v[n]=numbpart(n) + WeighT(v[1..n])[n]); v} \\ Andrew Howroyd, Oct 25 2018

Terms a(12) and beyond from Andrew Howroyd, Oct 25 2018

A320177 Number of series-reduced rooted identity trees whose leaves are strict integer partitions whose multiset union is an integer partition of n.

Original entry on oeis.org

1, 1, 3, 5, 11, 26, 65, 169, 463, 1294, 3691, 10700, 31417, 93175, 278805, 840424, 2549895, 7780472, 23860359, 73500838, 227330605, 705669634, 2197750615, 6865335389, 21505105039, 67533738479, 212575923471, 670572120240, 2119568530289, 6712115439347

Offset: 1

Gus Wiseman, Oct 07 2018

nonn

A rooted tree is series-reduced if every non-leaf node has at least two branches.

In an identity tree, all branches directly under any given node are different.

			The a(1) = 1 through a(5) = 11 rooted trees:
  (1)  (2)  (3)       (4)            (5)
            (21)      (31)           (32)
            ((1)(2))  ((1)(3))       (41)
                      ((1)(12))      ((1)(4))
                      ((1)((1)(2)))  ((2)(3))
                                     ((1)(13))
                                     ((2)(12))
                                     ((1)((1)(3)))
                                     ((2)((1)(2)))
                                     ((1)((1)(12)))
                                     ((1)((1)((1)(2))))

Andrew Howroyd, Table of n, a(n) for n = 1..200

Cf. A000669, A004111, A005804, A141268, A292504, A300660, A319312.

Cf. A320171, A320174, A320175, A320176, A320178.

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]]]];
gog[m_]:=If[UnsameQ@@m,Prepend[#,m],#]&[Join@@Table[Select[Union[Sort/@Tuples[gog/@p]],UnsameQ@@#&],{p,Select[mps[m],Length[#]>1&]}]];
Table[Length[Join@@Table[gog[m],{m,IntegerPartitions[n]}]],{n,10}]

WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}
seq(n)={my(p=prod(k=1, n, 1 + x^k + O(x*x^n)), v=vector(n)); for(n=1, n, v[n]=polcoef(p, n) + WeighT(v[1..n])[n]); v} \\ Andrew Howroyd, Oct 25 2018

Terms a(13) and beyond from Andrew Howroyd, Oct 25 2018

A320178 Number of series-reduced rooted identity trees whose leaves are constant integer partitions whose multiset union is an integer partition of n.

Original entry on oeis.org

1, 2, 4, 8, 19, 53, 151, 459, 1445, 4634, 15154, 50253, 168607, 571212, 1951588, 6715575, 23255444, 80978697, 283373024, 995995996, 3514614634, 12446666967, 44222390525, 157587392768, 563096832839, 2017121728223, 7242436444030, 26059512879605, 93952946906117

Offset: 1

Gus Wiseman, Oct 07 2018

nonn

A rooted tree is series-reduced if every non-leaf node has at least two branches.

In an identity tree, all branches directly under any given node are different.

			The a(1) = 1 through a(5) = 19 rooted trees:
  (1)  (2)   (3)        (4)             (5)
       (11)  (111)      (22)            (11111)
             ((1)(2))   (1111)          ((1)(4))
             ((1)(11))  ((1)(3))        ((2)(3))
                        ((2)(11))       ((1)(22))
                        ((1)(111))      ((3)(11))
                        ((1)((1)(2)))   ((2)(111))
                        ((1)((1)(11)))  ((1)(1111))
                                        ((11)(111))
                                        ((1)(2)(11))
                                        ((1)((1)(3)))
                                        ((2)((1)(2)))
                                        ((11)((1)(2)))
                                        ((1)((2)(11)))
                                        ((2)((1)(11)))
                                        ((1)((1)(111)))
                                        ((11)((1)(11)))
                                        ((1)((1)((1)(2))))
                                        ((1)((1)((1)(11))))

Andrew Howroyd, Table of n, a(n) for n = 1..200

Cf. A000669, A004111, A005804, A141268, A292504, A300660, A319312.

Cf. A320171, A320174, A320175, A320176, A320177.

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]]]];
gob[m_]:=If[SameQ@@m,Prepend[#,m],#]&[Join@@Table[Select[Union[Sort/@Tuples[gob/@p]],UnsameQ@@#&],{p,Select[mps[m],Length[#]>1&]}]];
Table[Length[Join@@Table[gob[m],{m,IntegerPartitions[n]}]],{n,10}]

WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}
seq(n)={my(v=vector(n)); for(n=1, n, v[n]=numdiv(n) + WeighT(v[1..n])[n]); v} \\ Andrew Howroyd, Oct 25 2018

Terms a(13) and beyond from Andrew Howroyd, Oct 25 2018

A320289 Number of phylogenetic trees with n labels and no singleton leaves.

Original entry on oeis.org

0, 1, 1, 4, 11, 86, 477, 4810, 40679, 496522, 5662933, 81759910, 1169640551, 19622623190, 336215135973, 6455705990674, 128445712218263, 2785761076726066, 62980942321570981, 1525318051255683598, 38566041706375722071, 1032726237783455193662

Offset: 1

Gus Wiseman, Oct 09 2018

nonn

			The a(2) = 1 through a(5) = 11 phylogenetic trees:
  (12)  (123)  (1234)      (12345)
               ((12)(34))  ((12)(345))
               ((13)(24))  ((13)(245))
               ((14)(23))  ((14)(235))
                           ((15)(234))
                           ((23)(145))
                           ((24)(135))
                           ((25)(134))
                           ((34)(125))
                           ((35)(124))
                           ((45)(123))

Andrew Howroyd, Table of n, a(n) for n = 1..200

Cf. A000311, A000669, A002865, A005804, A141268, A300660, A304966, A304967, A319312, A320294, A320295.

numSetPtnsOfType[ptn_]:=Total[ptn]!/Times@@Factorial/@ptn/Times@@Factorial/@Length/@Split[ptn];
rotf[n_]:=rotf[n]=If[n==1,0,1+Sum[numSetPtnsOfType[p]*Times@@rotf/@p,{p,Select[IntegerPartitions[n],Length[#]>1&]}]];
Array[rotf,20]

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
b(n,k)={my(v=vector(n)); for(n=2, n, v[n]=binomial(n+k-1, n) + EulerT(v[1..n])[n]); v}
seq(n)={my(M=Mat(vectorv(n, k, b(n,k)))); vector(n, k, sum(i=1, k, binomial(k, i)*(-1)^(k-i)*M[i,k]))} \\ Andrew Howroyd, Oct 26 2018

A320295 Number of series-reduced rooted trees whose leaves are non-singleton integer partitions whose multiset union is an integer partition of n.

Original entry on oeis.org

0, 1, 2, 5, 8, 19, 34, 80, 165, 394, 892, 2192, 5232, 13057, 32271, 81568, 205748, 525735, 1344828, 3467415, 8960849, 23280323, 60639680, 158559047, 415631368, 1092734050, 2879420753, 7605713020, 20130266302, 53386744298, 141836904569, 377479973474, 1006189769886

Offset: 1

Gus Wiseman, Oct 09 2018

nonn

Also phylogenetic trees with no singleton leaves on integer partitions of n.

			The a(2) = 1 through a(6) = 19 trees:
  (11)  (21)   (22)        (32)         (33)
        (111)  (31)        (41)         (42)
               (211)       (221)        (51)
               (1111)      (311)        (222)
               ((11)(11))  (2111)       (321)
                           (11111)      (411)
                           ((11)(12))   (2211)
                           ((11)(111))  (3111)
                                        (21111)
                                        (111111)
                                        ((11)(13))
                                        ((11)(22))
                                        ((12)(12))
                                        ((11)(112))
                                        ((12)(111))
                                        ((11)(1111))
                                        ((111)(111))
                                        ((11)(11)(11))
                                        ((11)((11)(11)))

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

Cf. A000311, A000669, A005804, A141268, A302545, A304966, A319312, A320289, A320294.

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]]]];
pgtm[m_]:=Prepend[Join@@Table[Union[Sort/@Tuples[pgtm/@p]],{p,Select[mps[m],Length[#]>1&]}],m];
Table[Sum[Length[Select[pgtm[m],FreeQ[#,{_}]&]],{m,IntegerPartitions[n]}],{n,14}]

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(p=1/prod(k=1, n, 1 - x^k + O(x*x^n)), v=vector(n)); for(n=1, n, v[n]=polcoef(p, n) - 1 + EulerT(v[1..n])[n]); v} \\ Andrew Howroyd, Oct 25 2018

Terms a(12) and beyond from Andrew Howroyd, Oct 25 2018

A320296 Number of series-reduced rooted trees whose leaves form an integer partition of n with no 1's.

Original entry on oeis.org

0, 1, 1, 2, 2, 5, 6, 15, 22, 51, 86, 195, 354, 781, 1512, 3286, 6602, 14269, 29424, 63494, 133298, 287909, 612188, 1325375, 2844448, 6176145, 13348858, 29074164, 63187176, 138044144, 301350424, 660265471, 1446678326, 3178246273, 6985464590, 15384556290

Offset: 1

Gus Wiseman, Oct 09 2018

nonn

Also phylogenetic trees with n unlabeled objects and no singleton leaves.

			The a(2) = 1 through a(9) = 22 trees:
2   3   4     5     6        7        8           9
        (22)  (23)  (24)     (25)     (26)        (27)
                    (33)     (34)     (35)        (36)
                    (222)    (223)    (44)        (45)
                    (2(22))  ((22)3)  (224)       (225)
                             (2(23))  (233)       (234)
                                      (2222)      (333)
                                      ((22)4)     (2223)
                                      (2(24))     ((22)5)
                                      ((23)3)     (2(25))
                                      (2(33))     ((23)4)
                                      (2(222))    (2(34))
                                      (22(22))    ((24)3)
                                      ((22)(22))  ((33)3)
                                      (2(2(22)))  (2(22)3)
                                                  (2(223))
                                                  (22(23))
                                                  (3(222))
                                                  ((2(22))3)
                                                  ((22)(23))
                                                  (2((22)3))
                                                  (2(2(23)))

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

Cf. A000311, A000669, A002865, A005804, A141268, A304967, A319312, A320293.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
t[n_]:=t[n]=If[PrimeQ[n],{n},Join@@Table[Union[Sort/@Tuples[t/@fac]],{fac,Select[facs[n],Length[#]>1&]}]];
Table[Sum[Length[t[Times@@Prime/@ptn]],{ptn,Select[IntegerPartitions[n],FreeQ[#,1]&]}],{n,15}]

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(v=vector(n)); for(n=2, n, v[n]=1 + EulerT(v[1..n])[n]); v} \\ Andrew Howroyd, Oct 25 2018

Terms a(26) and beyond from Andrew Howroyd, Oct 25 2018

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}]

cp's OEIS Frontend

A320175 Number of series-reduced rooted trees whose leaves are strict integer partitions whose multiset union is an integer partition of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A330668 Number of non-isomorphic balanced reduced multisystems of weight n whose leaves (which are multisets of atoms) are all sets.

Views

Author

Keywords

Comments

Examples

Crossrefs

A320171 Number of series-reduced rooted identity trees whose leaves are integer partitions whose multiset union is an integer partition of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A320177 Number of series-reduced rooted identity trees whose leaves are strict integer partitions whose multiset union is an integer partition of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A320178 Number of series-reduced rooted identity trees whose leaves are constant integer partitions whose multiset union is an integer partition of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A320289 Number of phylogenetic trees with n labels and no singleton leaves.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A320295 Number of series-reduced rooted trees whose leaves are non-singleton integer partitions whose multiset union is an integer partition of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A320296 Number of series-reduced rooted trees whose leaves form an integer partition of n with no 1's.

Views