A292504 -id:A292504 - cp's OEIS frontend

A002033 Number of perfect partitions of n.

1, 1, 1, 2, 1, 3, 1, 4, 2, 3, 1, 8, 1, 3, 3, 8, 1, 8, 1, 8, 3, 3, 1, 20, 2, 3, 4, 8, 1, 13, 1, 16, 3, 3, 3, 26, 1, 3, 3, 20, 1, 13, 1, 8, 8, 3, 1, 48, 2, 8, 3, 8, 1, 20, 3, 20, 3, 3, 1, 44, 1, 3, 8, 32, 3, 13, 1, 8, 3, 13, 1, 76, 1, 3, 8, 8, 3, 13, 1, 48, 8, 3, 1, 44, 3, 3, 3, 20, 1, 44, 3, 8, 3, 3, 3, 112

Offset: 0

N. J. A. Sloane

A perfect partition of n is one which contains just one partition of every number less than n when repeated parts are regarded as indistinguishable. Thus 1^n is a perfect partition for every n; and for n = 7, 4 1^3, 4 2 1, 2^3 1 and 1^7 are all perfect partitions. [Riordan]
Also number of ordered factorizations of n+1, see A074206.
Also number of gozinta chains from 1 to n (see A034776). - David W. Wilson
a(n) is the permanent of the n X n matrix with (i,j) entry = 1 if j|i+1 and = 0 otherwise. For n=3 the matrix is {{1, 1, 0}, {1, 0, 1}, {1, 1, 0}} with permanent = 2. - David Callan, Oct 19 2005
Appears to be the number of permutations that contribute to the determinant that gives the Moebius function. Verified up to a(9). - Mats Granvik, Sep 13 2008
Dirichlet inverse of A153881 (assuming offset 1). - Mats Granvik, Jan 03 2009
Equals row sums of triangle A176917. - Gary W. Adamson, Apr 28 2010
A partition is perfect iff it is complete (A126796) and knapsack (A108917). - Gus Wiseman, Jun 22 2016
a(n) is also the number of series-reduced planted achiral trees with n + 1 unlabeled leaves, where a rooted tree is series-reduced if all terminal subtrees have at least two branches, and achiral if all branches directly under any given node are equal. Also Moebius transform of A067824. - Gus Wiseman, Jul 13 2018

			n=0: 1 (the empty partition)
n=1: 1 (1)
n=2: 1 (11)
n=3: 2 (21, 111)
n=4: 1 (1111)
n=5: 3 (311, 221, 11111)
n=6: 1 (111111)
n=7: 4 (4111, 421, 2221, 1111111)
From _Gus Wiseman_, Jul 13 2018: (Start)
The a(11) = 8 series-reduced planted achiral trees with 12 unlabeled leaves:
  (oooooooooooo)
  ((oooooo)(oooooo))
  ((oooo)(oooo)(oooo))
  ((ooo)(ooo)(ooo)(ooo))
  ((oo)(oo)(oo)(oo)(oo)(oo))
  (((ooo)(ooo))((ooo)(ooo)))
  (((oo)(oo)(oo))((oo)(oo)(oo)))
  (((oo)(oo))((oo)(oo))((oo)(oo)))
(End)

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 126, see #27.
R. Honsberger, Mathematical Gems III, M.A.A., 1985, p. 141.
D. E. Knuth, The Art of Computer Programming, Pre-Fasc. 3b, Sect. 7.2.1.5, no. 67, p. 25.
P. A. MacMahon, The theory of perfect partitions and the compositions of multipartite numbers, Messenger Math., 20 (1891), 103-119.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 123-124.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..9999
Daniele A. Gewurz and Francesca Merola, Sequences realized as Parker vectors of oligomorphic permutation groups, J. Integer Seqs., Vol. 6, 2003.
HoKyu Lee, Double perfect partitions, Discrete Math., 306 (2006), 519-525.
Paul Pollack, On the parity of the number of multiplicative partitions and related problems, Proc. Amer. Math. Soc. 140 (2012), 3793-3803.
J. Riordan, Letter to N. J. A. Sloane, Dec. 1970
Eric Weisstein's World of Mathematics, Perfect Partition
Eric Weisstein's World of Mathematics, Dirichlet Series Generating Function
Index entries for "core" sequences

Same as A074206, up to the offset and initial term there.
Cf. A001055, A050324.
a(A002110)=A000670.
Cf. A000123, A100529, A117621.
Cf. A176917.
For parity see A008966.
Cf. A126796, A108917.
Cf. A001678, A003238, A067824, A167865, A214577, A289078, A292504, A316782.

a := array(1..150): for k from 1 to 150 do a[k] := 0 od: a[1] := 1: for j from 2 to 150 do for m from 1 to j-1 do if j mod m = 0 then a[j] := a[j]+a[m] fi: od: od: for k from 1 to 150 do printf(`%d,`,a[k]) od: # James Sellers, Dec 07 2000
# alternative
A002033 := proc(n)
    option remember;
    local a;
    if n <= 2 then
        return 1;
    else
        a := 0 ;
        for i from 0 to n-1 do
            if modp(n+1,i+1) = 0 then
                a := a+procname(i);
            end if;
        end do:
    end if;
    a ;
end proc: # R. J. Mathar, May 25 2017

a[0] = 1; a[1] = 1; a[n_] := a[n] = a /@ Most[Divisors[n]] // Total; a /@ Range[96]  (* Jean-François Alcover, Apr 06 2011, updated Sep 23 2014. NOTE: This produces A074206(n) = a(n-1). - M. F. Hasler, Oct 12 2018 *)

A002033(n) = if(n,sumdiv(n+1,i,if(i<=n,A002033(i-1))),1) \\ Michael B. Porter, Nov 01 2009, corrected by M. F. Hasler, Oct 12 2018

from functools import lru_cache
from sympy import divisors
@lru_cache(maxsize=None)
def A002033(n):
    if n <= 1:
        return 1
    return sum(A002033(i-1) for i in divisors(n+1,generator=True) if i <= n) # Chai Wah Wu, Jan 12 2022

From David Wasserman, Nov 14 2006: (Start)
a(n-1) = Sum_{i|d, i < n} a(i-1).
a(p^k-1) = 2^(k-1).
a(n-1) = A067824(n)/2 for n > 1.
a(A122408(n)-1) = A122408(n)/2. (End)
a(A025487(n)-1) = A050324(n). - R. J. Mathar, May 26 2017
a(n) = (A253249(n+1)+1)/4, n > 0. - Geoffrey Critzer, Aug 19 2020

Edited by M. F. Hasler, Oct 12 2018

A141268 Number of phylogenetic rooted trees with n unlabeled objects.

Original entry on oeis.org

1, 2, 4, 11, 30, 96, 308, 1052, 3648, 13003, 47006, 172605, 640662, 2402388, 9082538, 34590673, 132566826, 510904724, 1978728356, 7697565819, 30063818314, 117840547815, 463405921002, 1827768388175, 7228779397588, 28661434308095, 113903170011006, 453632267633931

Offset: 1

Thomas Wieder, Jun 20 2008

nonn

Unlabeled analog of A005804 = Phylogenetic trees with n labels.
From Gus Wiseman, Jul 31 2018: (Start)
a(n) is the number of series-reduced rooted trees whose leaves form an integer partition of n. For example, the following are the a(4) = 11 series-reduced rooted trees whose leaves form an integer partition of 4.
  4,
  (13),
  (22),
  (112), (1(12)), (2(11)),
  (1111), (11(11)), (1(1(11))), (1(111)), ((11)(11)).
(End)

			For n=4 we have A141268(4)=11 because
Set(Set(Z),Set(Z),Set(Z,Z)),
Set(Set(Z),Set(Set(Z),Set(Z,Z))),
Set(Z,Z,Z,Z),
Set(Set(Z,Z),Set(Z,Z)),
Set(Set(Set(Z),Set(Z)),Set(Z,Z)),
Set(Set(Z),Set(Z),Set(Set(Z),Set(Z))),
Set(Set(Z),Set(Z),Set(Z),Set(Z)),
Set(Set(Z),Set(Set(Z),Set(Z),Set(Z))),
Set(Set(Set(Z),Set(Z)),Set(Set(Z),Set(Z))),
Set(Set(Z),Set(Z,Z,Z)),
Set(Set(Z),Set(Set(Z),Set(Set(Z),Set(Z))))

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Moshe Klein and A. Yu. Khrennikov, Recursion over partitions, P-Adic Numbers, Ultrametric Analysis, and Applications 6.4 (2014): 303-309. See sp_n.

Cf. A000081, A000311, A000669, A001678, A005804, A141268, A292504, A300660, A316655.

with(combstruct): A141268 := [H, {H=Union(Set(Z,card>=1),Set(H,card>=2))}, unlabelled]; seq(count(A141268, size=j), j=1..20);
# second Maple program:
b:= proc(n,i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1)*binomial(a(i)+j-1, j), j=0..n/i)))
    end:
a:= n-> `if`(n<2, n, 1+b(n, n-1)):
seq(a(n), n=1..30);  # Alois P. Heinz, Jun 18 2018

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,IntegerPartitions[n]}],{n,7}] (* Gus Wiseman, Jul 31 2018 *)
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0,
     Sum[b[n-i*j, i-1]*Binomial[a[i]+j-1, j], {j, 0, n/i}]]];
a[n_] := If[n < 2, n, 1 + b[n, n-1]];
Array[a, 30] (* Jean-François Alcover, May 21 2021, after Alois P. Heinz *)

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(v=vector(n)); for(n=1, n, v[n]=1 + EulerT(v[1..n])[n]); v} \\ Andrew Howroyd, Oct 26 2018

a(n) ~ c * d^n / n^(3/2), where d = 4.210216501727104448901818751..., c = 0.21649387167268793159311306... . - Vaclav Kotesovec, Sep 04 2014

Offset corrected and more terms from Alois P. Heinz, Apr 21 2012

A005804 Number of phylogenetic rooted trees with n labels.

Original entry on oeis.org

1, 2, 8, 58, 612, 8374, 140408, 2785906, 63830764, 1658336270, 48169385024, 1546832023114, 54413083601268, 2080827594898342, 85948745163598088, 3813417859420469410, 180876816831806597500, 9133309115320844870078, 489156459621633161274704, 27696066472039561313329018

Offset: 1

N. J. A. Sloane

These are series-reduced rooted trees where each leaf is a nonempty subset of the set of n labels.

See A141268 for phylogenetic rooted trees with n unlabeled objects. - Thomas Wieder, Jun 20 2008

			a(3)=8 because we have:
  Set(Set(Z[3]),Set(Z[1]),Set(Z[2])),
  Set(Z[3],Z[2],Z[1]),
  Set(Set(Z[3],Z[1]),Set(Z[2])),
  Set(Set(Set(Z[3]),Set(Z[2])),Set(Z[1])),
  Set(Set(Set(Z[3]),Set(Z[1])),Set(Z[2])),
  Set(Set(Z[3]),Set(Set(Z[1]),Set(Z[2]))),
  Set(Set(Z[3]),Set(Z[2],Z[1])),
  Set(Set(Z[3],Z[2]),Set(Z[1])).
From _Gus Wiseman_, Jul 31 2018: (Start)
The 8 series-reduced rooted trees whose leaves are a set partition of {1,2,3}:
  {1,2,3}
  ({1}{2,3})
  ({1}({2}{3}))
  ({2}{1,3})
  ({2}({1}{3}))
  ({3}{1,2})
  ({3}({1}{2}))
  ({1}{2}{3})
(End)

Foulds, L. R.; Robinson, R. W. Enumeration of phylogenetic trees without points of degree two. Ars Combin. 17 (1984), A, 169-183. Math. Rev. 85f:05045
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 1..380 (first 100 terms from T. D. Noe)
N. J. A. Sloane, Transforms
Index entries for sequences related to rooted trees

Cf. A000081, A000311, A000669, A001678, A005805, A141268, A292504, A300660, A316656.

# From Thomas Wieder, Jun 20 2008: (Start)
ser := series(-LambertW(-1/2*exp(1/2*exp(z)-1)) + 1/2*exp(z)-1, z=0, 10);
seq(n!*coeff(ser, z, n), n = 1..9);
# Alternative:
with(combstruct):
A005804 := [H, {H=Union(Set(Z,card>=1), Set(H,card>=2))}, labelled];
seq(count(A005804,size=j), j=1..20);
# (End)

numSetPtnsOfType[ptn_]:=Total[ptn]!/Times@@Factorial/@ptn/Times@@Factorial/@Length/@Split[ptn];
a[n_]:=a[n]=If[n==1,1,1+Sum[numSetPtnsOfType[ptn]*Times@@a/@ptn,{ptn,Rest[IntegerPartitions[n]]}]];
Array[a,20] (* Gus Wiseman, Jul 31 2018 *)

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
b(n,k)={my(v=vector(n)); for(n=1, 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

Stirling transform of [ 1, 1, 4, 26, 236, ... ] = A000311 [ Foulds and Robinson ].
E.g.f.: -LambertW(-(1/2)*exp((1/2)*exp(z) - 1)) + (1/2)*exp(z) - 1. - Thomas Wieder, Jun 20 2008
a(n) ~ sqrt(log(2))*(log(2)+log(log(2)))^(1/2-n)*n^(n-1)/exp(n). - Vaclav Kotesovec, Aug 07 2013
E.g.f. f(x) satisfies  2*f(x) - exp(f(x)) = exp(x) - 2. - Gus Wiseman, Jul 31 2018

More terms, comment from Christian G. Bower, Dec 15 1999

A339645 Triangle read by rows: T(n,k) is the number of inequivalent colorings of lone-child-avoiding rooted trees with n colored leaves using exactly k colors.

Original entry on oeis.org

1, 1, 1, 2, 3, 2, 5, 17, 12, 5, 12, 73, 95, 44, 12, 33, 369, 721, 512, 168, 33, 90, 1795, 5487, 5480, 2556, 625, 90, 261, 9192, 41945, 58990, 36711, 12306, 2342, 261, 766, 47324, 321951, 625088, 516952, 224241, 57155, 8702, 766, 2312, 249164, 2483192, 6593103, 7141755, 3965673, 1283624, 258887, 32313, 2312

Offset: 1

Andrew Howroyd, Dec 11 2020

Only the leaves are colored. Equivalence is up to permutation of the colors.

Lone-child-avoiding rooted trees are also called planted series-reduced trees in some other sequences.

			Triangle begins:
    1;
    1,     1;
    2,     3,      2;
    5,    17,     12,      5;
   12,    73,     95,     44,     12;
   33,   369,    721,    512,    168,     33;
   90,  1795,   5487,   5480,   2556,    625,    90;
  261,  9192,  41945,  58990,  36711,  12306,  2342,  261;
  766, 47324, 321951, 625088, 516952, 224241, 57155, 8702, 766;
  ...
From _Gus Wiseman_, Jan 02 2021: (Start)
Non-isomorphic representatives of the 39 = 5 + 17 + 12 + 5 trees with four colored leaves:
  (1111)      (1112)      (1123)      (1234)
  (1(111))    (1122)      (1(123))    (1(234))
  (11(11))    (1(112))    (11(23))    (12(34))
  ((11)(11))  (11(12))    (12(13))    ((12)(34))
  (1(1(11)))  (1(122))    (2(113))    (1(2(34)))
              (11(22))    (23(11))
              (12(11))    ((11)(23))
              (12(12))    (1(1(23)))
              (2(111))    ((12)(13))
              ((11)(12))  (1(2(13)))
              (1(1(12)))  (2(1(13)))
              ((11)(22))  (2(3(11)))
              (1(1(22)))
              (1(2(11)))
              ((12)(12))
              (1(2(12)))
              (2(1(11)))
(End)

Andrew Howroyd, PARI Functions for Combinatorial Species, v2, Dec 2020.
Wikipedia, Combinatorial species

The case with only one color is A000669.
Counting by nodes gives A318231.
A labeled version is A319376.
Row sums are A330470.
A000311 counts singleton-reduced phylogenetic trees.
A001678 counts unlabeled lone-child-avoiding rooted trees.
A005121 counts chains of set partitions, with maximal case A002846.
A005804 counts phylogenetic rooted trees with n labels.
A060356 counts labeled lone-child-avoiding rooted trees.
A141268 counts lone-child-avoiding rooted trees with leaves summing to n.
A291636 lists Matula-Goebel numbers of lone-child-avoiding rooted trees.
A316651 counts lone-child-avoiding rooted trees with normal leaves.
A316652 counts lone-child-avoiding rooted trees with strongly normal leaves.
A330465 counts inequivalent leaf-colorings of phylogenetic rooted trees.
Cf. A196545, A213427, A281118, A289501, A292504, A318812, A319312, A330627.

\\ See link above for combinatorial species functions.
cycleIndexSeries(n)={my(v=vector(n)); v[1]=sv(1); for(n=2, #v, v[n] = polcoef( sExp(x*Ser(v[1..n])), n )); x*Ser(v)}
{my(A=InequivalentColoringsTriangle(cycleIndexSeries(10))); for(n=1, #A~, print(A[n,1..n]))}

A300660 Number of unlabeled rooted phylogenetic trees with n (leaf-) nodes such that for each inner node all children are either leaves or roots of distinct subtrees.

Original entry on oeis.org

0, 1, 1, 2, 3, 6, 13, 30, 72, 182, 467, 1222, 3245, 8722, 23663, 64758, 178459, 494922, 1380105, 3867414, 10884821, 30756410, 87215419, 248117618, 707952902, 2025479210, 5809424605, 16700811214, 48113496645, 138884979562, 401645917999, 1163530868090

Offset: 0

Alois P. Heinz, Jun 18 2018

From Gus Wiseman, Jul 31 2018 and Feb 06 2020: (Start)
a(n) is the number of lone-child-avoiding rooted identity trees whose leaves form an integer partition of n. For example, the following are the a(6) = 13 lone-child-avoiding rooted identity trees whose leaves form an integer partition of 6.
  6,
  (15),
  (24),
  (123), (1(23)), (2(13)), (3(12)),
  (1(14)),
  (1(1(13))),
  (12(12)), (1(2(12))), (2(1(12))),
  (1(1(1(12)))).
(End)

			:   a(3) = 2:        :   a(4) = 3:                      :
:      o       o     :        o         o        o      :
:     / \     /|\    :       / \       / \     /( )\    :
:    o   N   N N N   :      o   N     o   N   N N N N   :
:   ( )              :     / \       /|\                :
:   N N              :    o   N     N N N               :
:                    :   ( )                            :
:                    :   N N                            :
From _Gus Wiseman_, Feb 06 2020: (Start)
The a(2) = 1 through a(6) = 13 unlabeled rooted phylogenetic semi-identity trees:
  (oo) (ooo)     (oooo)         (ooooo)             (oooooo)
       ((o)(oo)) ((o)(ooo))     ((o)(oooo))         ((o)(ooooo))
                 ((o)((o)(oo))) ((oo)(ooo))         ((oo)(oooo))
                                ((o)((o)(ooo)))     ((o)(oo)(ooo))
                                ((oo)((o)(oo)))     (((o)(oo))(ooo))
                                ((o)((o)((o)(oo)))) ((o)((o)(oooo)))
                                                    ((o)((oo)(ooo)))
                                                    ((oo)((o)(ooo)))
                                                    ((o)(oo)((o)(oo)))
                                                    ((o)((o)((o)(ooo))))
                                                    ((o)((oo)((o)(oo))))
                                                    ((oo)((o)((o)(oo))))
                                                    ((o)((o)((o)((o)(oo)))))
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..2079
Wikipedia, Phylogenetic tree
Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.
Index entries for sequences related to rooted trees
Index entries for sequences related to trees

Cf. A000081, A004111, A141268, A289501, A301467.
Cf. A000669, A001678, A005804, A292504, A300660, A316653, A316654, A316656.
The locally disjoint case is A316694.
Cf. A276625, A306200, A319312, A331679, A331686, A331875.

b:= proc(n,i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1)*binomial(a(i), j), j=0..n/i)))
    end:
a:= n-> `if`(n=0, 0, 1+b(n, n-1)):
seq(a(n), n=0..30);

b[0, ] = 1; b[, _?NonPositive] = 0;
b[n_, i_] := b[n, i] = Sum[b[n-i*j, i-1]*Binomial[a[i], j], {j, 0, n/i}];
a[0] = 0; a[n_] := a[n] = 1 + b[n, n-1];
Table[a[n], {n, 0, 31}] (* Jean-François Alcover, May 03 2019, from Maple *)
ursit[n_]:=Prepend[Join@@Table[Select[Union[Sort/@Tuples[ursit/@ptn]],UnsameQ@@#&],{ptn,Select[IntegerPartitions[n],Length[#]>1&]}],n];
Table[Length[ursit[n]],{n,10}] (* Gus Wiseman, Feb 06 2020 *)

a(n) ~ c * d^n / n^(3/2), where d = 3.045141208159736483720243229947630323380565686... and c = 0.2004129296838557718008171812000512670126... - Vaclav Kotesovec, Aug 27 2018

A319312 Number of series-reduced rooted trees whose leaves are integer partitions whose multiset union is an integer partition of n.

Original entry on oeis.org

1, 3, 7, 22, 67, 242, 885, 3456, 13761, 56342, 234269, 989335, 4225341, 18231145, 79321931, 347676128, 1533613723, 6803017863, 30328303589, 135808891308, 610582497919, 2755053631909, 12472134557093, 56630659451541, 257841726747551, 1176927093597201

Offset: 1

Gus Wiseman, Sep 17 2018

nonn

Also the number of orderless tree-factorizations of Heinz numbers of integer partitions of n.

Also the number of phylogenetic trees on a multiset of labels summing to n.

			The a(3) = 7 trees:
  (3)    (21)        (111)
       ((1)(2))    ((1)(11))
                  ((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, A316653, A316654, A316656.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
phyfacs[n_]:=Prepend[Join@@Table[Union[Sort/@Tuples[phyfacs/@f]],{f,Select[facs[n],Length[#]>1&]}],n];
Table[Sum[Length[phyfacs[Times@@Prime/@m]],{m,IntegerPartitions[n]}],{n,6}]

EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
seq(n)={my(v=[]); for(n=1, n, v=concat(v, numbpart(n) + EulerT(concat(v,[0]))[n])); v} \\ Andrew Howroyd, Sep 18 2018

Terms a(14) and beyond from Andrew Howroyd, Sep 18 2018

A060356 Expansion of e.g.f.: -LambertW(-x/(1+x)).

Original entry on oeis.org

0, 1, 0, 3, 4, 65, 306, 4207, 38424, 573057, 7753510, 134046671, 2353898196, 47602871329, 1013794852266, 23751106404495, 590663769125296, 15806094859299329, 448284980183376078, 13515502344669830287

Offset: 0

Vladeta Jovovic, Apr 01 2001

Also the number of labeled lone-child-avoiding rooted trees with n nodes. A rooted tree is lone-child-avoiding if it has no unary branchings, meaning every non-leaf node covers at least two other nodes. The unlabeled version is A001678(n + 1). - Gus Wiseman, Jan 20 2020

			From _Gus Wiseman_, Dec 31 2019: (Start)
Non-isomorphic representatives of the a(7) = 4207 trees, written as root[branches], are:
  1[2,3[4,5[6,7]]]
  1[2,3[4,5,6,7]]
  1[2[3,4],5[6,7]]
  1[2,3,4[5,6,7]]
  1[2,3,4,5[6,7]]
  1[2,3,4,5,6,7]
(End)

Harry J. Smith, Table of n, a(n) for n = 0..100
David Callan, A sign-reversing involution to count labeled lone-child-avoiding trees, arXiv:1406.7784 [math.CO], (30-June-2014).
Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.

Cf. A052871, A060313.
Cf. A008297.
Column k=0 of A231602.
The unlabeled version is A001678(n + 1).
The case where the root is fixed is A108919.
Unlabeled rooted trees are counted by A000081.
Lone-child-avoiding rooted trees with labeled leaves are A000311.
Matula-Goebel numbers of lone-child-avoiding rooted trees are A291636.
Singleton-reduced rooted trees are counted by A330951.
Cf. A000669, A004111, A005121, A048816, A292504, A316651, A316652, A318231, A318813, A330465, A330624.

List([0..20],n->Sum([1..n],k->(-1)^(n-k)*Factorial(n)/Factorial(k) *Binomial(n-1,k-1)*k^(k-1))); # Muniru A Asiru, Feb 19 2018

seq(coeff(series( -LambertW(-x/(1+x)), x, n+1), x, n)*n!, n = 0..20); # G. C. Greubel, Mar 16 2020

CoefficientList[Series[-LambertW[-x/(1+x)], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Nov 27 2012 *)
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
a[n_]:=If[n==1,1,n*Sum[Times@@a/@Length/@stn,{stn,Select[sps[Range[n-1]],Length[#]>1&]}]];
Array[a,10] (* Gus Wiseman, Dec 31 2019 *)

{ for (n=0, 100, f=n!; a=sum(k=1, n, (-1)^(n - k)*f/k!*binomial(n - 1, k - 1)*k^(k - 1)); write("b060356.txt", n, " ", a); ) } \\ Harry J. Smith, Jul 04 2009

my(x='x+O('x^20)); concat([0], Vec(serlaplace(-lambertw(-x/(1+x))))) \\ G. C. Greubel, Feb 19 2018

a(n) = Sum_{k=1..n} (-1)^(n-k)*n!/k!*binomial(n-1, k-1)*k^(k-1). a(n) = Sum_{k=0..n} Stirling1(n, k)*A058863(k). - Vladeta Jovovic, Sep 17 2003
a(n) ~ n^(n-1) * (1-exp(-1))^(n+1/2). - Vaclav Kotesovec, Nov 27 2012
a(n) = n * A108919(n). - Gus Wiseman, Dec 31 2019

A316651 Number of series-reduced rooted trees with n leaves spanning an initial interval of positive integers.

Original entry on oeis.org

1, 2, 12, 112, 1444, 24086, 492284, 11910790, 332827136, 10546558146, 373661603588, 14636326974270, 628032444609396, 29296137817622902, 1476092246351259964, 79889766016415899270, 4622371378514020301740, 284719443038735430679268, 18601385258191195218790756

Offset: 1

Gus Wiseman, Jul 09 2018

nonn

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

			The a(3) = 12 trees:
  (1(11)), (111),
  (1(12)), (2(11)), (112),
  (1(22)), (2(12)), (122),
  (1(23)), (2(13)), (3(12)), (123).

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

Cf. A000081, A000311, A000669, A001678, A005804, A034691, A141268, A292504.
Cf. A316652, A316653, A316654, A316655, A316656, A319369.
Row sums of A319376.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(A(i, k)+j-1, j)*b(n-i*j, i-1, k), j=0..n/i)))
    end:
A:= (n, k)-> `if`(n<2, n*k, b(n, n-1, k)):
a:= n-> add(add(A(n, k-j)*(-1)^j*binomial(k, j), j=0..k-1), k=1..n):
seq(a(n), n=1..20);  # Alois P. Heinz, Sep 18 2018

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]]]];
gro[m_]:=If[Length[m]==1,m,Union[Sort/@Join@@(Tuples[gro/@#]&/@Select[mps[m],Length[#]>1&])]];
allnorm[n_Integer]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
Table[Sum[Length[gro[m]],{m,allnorm[n]}],{n,5}]
(* Second program: *)
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0,
     Sum[Binomial[A[i, k] + j - 1, j] b[n - i*j, i - 1, k], {j, 0, n/i}]]];
A[n_, k_] := If[n < 2, n*k, b[n, n - 1, k]];
a[n_] := Sum[Sum[A[n, k-j]*(-1)^j*Binomial[k, j], {j, 0, k-1}], {k, 1, n}];
Array[a, 20] (* Jean-François Alcover, May 09 2021, after Alois P. Heinz *)

\\ here R(n,k) is A000669, A050381, A220823, ...
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
R(n,k)={my(v=[k]); for(n=2, n, v=concat(v, EulerT(concat(v,[0]))[n])); v}
seq(n)={sum(k=1, n, R(n,k)*sum(r=k, n, binomial(r,k)*(-1)^(r-k)) )} \\ Andrew Howroyd, Sep 14 2018

From Vaclav Kotesovec, Sep 18 2019: (Start)
a(n) ~ c * d^n * n^(n-1), where d = 1.37392076830840090205551979... and c = 0.41435722857311602982846...
a(n) ~ 2*log(2)*A326396(n)/n. (End)

Terms a(9) and beyond from Andrew Howroyd, Sep 14 2018

A316652 Number of series-reduced rooted trees whose leaves span an initial interval of positive integers with multiplicities an integer partition of n.

Original entry on oeis.org

1, 2, 9, 69, 623, 7793, 110430, 1906317, 36833614, 816101825, 19925210834, 541363267613, 15997458049946, 515769374925576, 17905023985615254, 669030297769291562, 26689471638523499483, 1134895275721374771655, 51161002326406795249910, 2440166138715867838359915

Offset: 1

Gus Wiseman, Jul 09 2018

nonn

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

			The a(3) = 9 trees:
(1(11)), (111),
(1(12)), (2(11)), (112),
(1(23)), (2(13)), (3(12)), (123).

Cf. A000081, A000311, A000669, A001678, A005804, A141268, A181821, A292504, A304660.

Cf. A316651, A316653, A316654, A316655, A316656.

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]]]];
gro[m_]:=If[Length[m]==1,m,Union[Sort/@Join@@(Tuples[gro/@#]&/@Select[mps[m],Length[#]>1&])]];
Table[Sum[Length[gro[m]],{m,Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n]}],{n,4}]

\\ See A339645 for combinatorial species functions.
cycleIndexSeries(n)={my(v=vector(n)); v[1]=sv(1); for(n=2, #v, v[n] = polcoef( sExp(x*Ser(v[1..n])), n )); x*Ser(v)}
StronglyNormalLabelingsSeq(cycleIndexSeries(15)) \\ Andrew Howroyd, Jan 04 2021

Terms a(10) and beyond from Andrew Howroyd, Jan 04 2021

A108919 Number of series-reduced labeled trees with n nodes.

Original entry on oeis.org

1, 0, 1, 1, 13, 51, 601, 4803, 63673, 775351, 12186061, 196158183, 3661759333, 72413918019, 1583407093633, 36916485570331, 929770285841137, 24904721121298671, 711342228666833173, 21502519995056598639, 687345492498807434461, 23135454269839313430715, 818568166383797223246601, 30357965273255025673685091

Offset: 1

Vladeta Jovovic, Jul 20 2005

"Series-reduced" means that if the tree is rooted at 1, then there is no node with just a single child.

Callan points out that A002792 is an incorrect version of this sequence. - Joerg Arndt, Jul 01 2014

Seiichi Manyama, Table of n, a(n) for n = 1..415
David Callan, A sign-reversing involution to count labeled lone-child-avoiding trees, arXiv:1406.7784 [math.CO], 2014.

The rooted version is A060356.

Cf. A000311, A000669, A001678, A292504, A316651, A316652, A318231, A318813, A330465, A330624, A352410.

f[n_] := Sum[(-1)^(n-k)*n!/k!*Binomial[n-1, k-1]*k^(k-1), {k, n}]/n; Table[ f[n], {n, 20}] (* Robert G. Wilson v, Jul 21 2005 *)

a(n) = { 1/n * sum(k=1, n, (-1)^(n-k) * binomial(n,k) * (n-1)!/(k-1)! * k^(k-1) ); } \\ Joerg Arndt, Aug 28 2014

a(n) = A060356(n)/n.
1 = Sum_{n>=0} a(n+1)*(exp(x)-x)^(-n-1)*x^n/n!.
E.g.f.: A(x) = Sum_{n>=0} a(n+1)*x^n/n! satisfies A(x) = exp(x*A(x))/(1+x). - Olivier Gérard, Dec 31 2013 (edited by Gus Wiseman, Dec 31 2019)
E.g.f.: -Integral (LambertW(-x/(1 + x))/x) dx. - Ilya Gutkovskiy, Jul 01 2020

More terms from Robert G. Wilson v, Jul 21 2005
New name (from A002792) by Joerg Arndt, Aug 28 2014
Offset corrected by Gus Wiseman, Dec 31 2019

cp's OEIS Frontend

A002033 Number of perfect partitions of n.

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A141268 Number of phylogenetic rooted trees with n unlabeled objects.

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A005804 Number of phylogenetic rooted trees with n labels.

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A339645 Triangle read by rows: T(n,k) is the number of inequivalent colorings of lone-child-avoiding rooted trees with n colored leaves using exactly k colors.

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A300660 Number of unlabeled rooted phylogenetic trees with n (leaf-) nodes such that for each inner node all children are either leaves or roots of distinct subtrees.

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A319312 Number of series-reduced rooted trees whose leaves are integer partitions whose multiset union is an integer partition of n.

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