A000311 -id:A000311 - cp's OEIS frontend

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.

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]))}

A048816 Number of rooted trees with n nodes with every leaf at the same height.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 12, 17, 28, 42, 68, 103, 168, 260, 420, 665, 1075, 1716, 2787, 4489, 7304, 11849, 19333, 31504, 51561, 84347, 138378, 227096, 373445, 614441, 1012583, 1669774, 2756951, 4555183, 7533988, 12469301, 20655523, 34238310, 56795325, 94270949

Offset: 1

Christian G. Bower, Apr 15 1999

nonn

The trees are unordered (see A000081). For balanced ordered rooted trees see A079500. - Joerg Arndt, Jul 20 2014

The trees are unlabeled. For labeled version see A238372. - Alois P. Heinz, Dec 29 2014

			See Arndt link.
From _Gus Wiseman_, Oct 08 2018: (Start)
The a(1) = 1 through a(7) = 12 balanced rooted trees with n nodes:
  o  (o)  (oo)   (ooo)    (oooo)     (ooooo)      (oooooo)
          ((o))  ((oo))   ((ooo))    ((oooo))     ((ooooo))
                 (((o)))  (((oo)))   (((ooo)))    (((oooo)))
                          ((o)(o))   ((o)(oo))    ((o)(ooo))
                          ((((o))))  ((((oo))))   ((oo)(oo))
                                     (((o)(o)))   ((((ooo))))
                                     (((((o)))))  (((o)(oo)))
                                                  ((o)(o)(o))
                                                  (((((oo)))))
                                                  ((((o)(o))))
                                                  (((o))((o)))
                                                  ((((((o))))))
(End)

Vaclav Kotesovec, Table of n, a(n) for n = 1..3500 (terms 1..300 from Alois P. Heinz)
Joerg Arndt, balanced unordered rooted trees for n = 1..10
Index entries for sequences related to rooted trees

Cf. A048808-A048815.
Row sums of A244925.
Cf. A079500, A238372.
Cf. A000081, A000311, A001678, A120803, A320154, A320160, A316624, A320169.

T[n_, k_] := T[n, k] = If[n==1, 1, If[k==0, 0, Sum[Sum[If[dJean-François Alcover, Jan 08 2016, after Alois P. Heinz *)

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

A053492 REVEGF transform of [1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, ...].

Original entry on oeis.org

1, 2, 15, 184, 3155, 69516, 1871583, 59542064, 2185497819, 90909876100, 4226300379983, 217152013181544, 12219893000227107, 747440554689309404, 49374719534173925055, 3503183373320829575008, 265693897270211120103563, 21451116469521758657525748

Offset: 1

N. J. A. Sloane, Jan 15 2000

nonn

Sequence gives the number of total circled partitions of n. This is the number of ways to partition n into at least two blocks, circle one block, then successively partition each non-singleton block into at least two blocks and circle one of the blocks. Stop when only singleton blocks remain. - Brian Drake, Apr 25 2006
a(n) is also the number of Schroeder trees on n vertices. - Brad R. Jones, May 09 2014
Number of pointed trees on pointed sets k[1...k...n] for any point k. - Gus Wiseman, Sep 27 2015

			E.g.f.: A(x) = x + 2*x^2/2! + 15*x^3/3! + 184*x^4/4! + 3155*x^5/5! + ...
Related expansions from _Paul D. Hanna_, Jul 07 2012: (Start)
A(x) = x + (exp(x)-1)*x + d/dx (exp(x)-1)^2*x^2/2! + d^2/dx^2 (exp(x)-1)^3*x^3/3! + d^3/dx^3 (exp(x)-1)^4*x^4/4! + ...
log(A(x)/x) = (exp(x)-1) + d/dx (exp(x)-1)^2*x/2! + d^2/dx^2 (exp(x)-1)^3*x^2/3! + d^3/dx^3 (exp(x)-1)^4*x^3/4! + ... (End)
The a(3) = 15 pointed trees are 1[1 2[2 3]], 1[1 3[2 3]], 1[1[1 3] 2], 1[1[1 2] 3], 1[1 2 3], 2[1 2[2 3]], 2[1[1 3] 2], 2[2 3[1 3]], 2[2[1 2] 3], 2[1 2 3], 3[1 3[2 3]], 3[2 3[1 3]], 3[1[1 2] 3], 3[2[1 2] 3], 3[1 2 3].

G. C. Greubel, Table of n, a(n) for n = 1..355
W. Y. Chen, A general bijective algorithm for trees, PNAS December 1, 1990 vol. 87 no. 24 9635-9639.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 854.
Elena L. Wang and Guoce Xin, On Ward Numbers and Increasing Schröder Trees, arXiv:2507.15654 [math.CO], 2025. See p. 12.

Cf. A000311, A029768, A052894, A262673.

A:= series(RootOf(exp(A053492:=%20n-%3E%20n!%20*%20coeff(A,%20x,%20n);%20%23%20_Brian%20Drake">Z)*_Z+x-2*_Z), x, 30): A053492:= n-> n! * coeff(A, x, n); # _Brian Drake, Apr 25 2006

Rest[CoefficientList[InverseSeries[Series[2*x-x*E^x, {x, 0, 20}], x],x] * Range[0, 20]!] (* Vaclav Kotesovec, Oct 27 2014 *)

a(n):= if n=1 then 1 else sum(k!*stirling2(n-1,k)*binomial(n+k-1,n-1),k,1,n-1); /* Vladimir Kruchinin, May 10 2011 */

{a(n) = if( n<1, 0, n! * polcoeff( serreverse( 2*x - x * exp(x + x * O(x^n))), n))}; /* Michael Somos, Jun 06 2012 */

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x); A=x+sum(m=1, n, Dx(m-1, (exp(x+x*O(x^n))-1)^m*x^m/m!)); n!*polcoeff(A, n)} \\ Paul D. Hanna, Jul 07 2012
for(n=1, 25, print1(a(n), ", "))

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); A=x*exp(sum(m=1, n, Dx(m-1, (exp(x+x*O(x^n))-1)^m*x^(m-1)/m!)+x*O(x^n))); n!*polcoeff(A, n)} \\ Paul D. Hanna, Jul 07 2012
for(n=1, 25, print1(a(n), ", "))

\p100 \\ set precision
{A=Vec(sum(n=0, 400, 1./(2 - n*x +O(x^25))^(n+1)) )}
for(n=1, #A, print1(round(A[n]), ", ")) \\ Paul D. Hanna, Oct 27 2014

E.g.f. is the compositional inverse of 2*x - x*exp(x). - Brian Drake, Apr 25 2006
E.g.f.: x + Sum_{n>=1} d^(n-1)/dx^(n-1) (exp(x)-1)^n*x^n / n!. - Paul D. Hanna, Jul 07 2012
E.g.f.: x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) (exp(x)-1)^n*x^(n-1) / n! ). - Paul D. Hanna, Jul 07 2012
a(n) = Sum_{k=1..n-1} k!*Stirling2(n-1,k)*C(n+k-1,n-1), n > 1, a(1)=1. - Vladimir Kruchinin, May 10 2011
O.g.f.: x*Sum_{n>=0}  1/(2 - n*x)^(n+1). - Paul D. Hanna, Oct 27 2014
a(n) ~ n^(n-1) * (LambertW(2*exp(1)))^n / (sqrt(1+LambertW(2*exp(1))) * 2^n * exp(n) * (LambertW(2*exp(1))-1)^(2*n-1)). - Vaclav Kotesovec, Oct 27 2014

Signs removed by Michael Somos, based on Brian Drake's remark, Jun 06 2012

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

A006024 Number of labeled mating graphs with n nodes. Also called point-determining graphs.

Original entry on oeis.org

1, 1, 1, 4, 32, 588, 21476, 1551368, 218608712, 60071657408, 32307552561088, 34179798520396032, 71474651351939175424, 296572048493274368856832, 2448649084251501449508762880, 40306353989748719650902623919616

Offset: 0

Simon Plouffe

nonn

A mating graph is one in which no two vertices have identical adjacencies with the other vertices. - Ronald C. Read and Vladeta Jovovic, Feb 10 2003

Also number of (n-1)-node labeled mating graphs allowing loops and without isolated nodes. - Vladeta Jovovic, Mar 08 2008

			Consider the square (cycle of length 4) on vertices 1, 2, 3 and 4 in that order. Join a fifth vertex (5) to vertices 1, 3 and 4. The resulting graph is not a mating graph since vertices 1 and 3 both have the set {2, 4, 5} as neighbors. If we delete the edge (1,5) then the resulting graph is a mating graph: the neighborhood sets for vertices 1, 2, 3, 4 and 5 are respectively {2,4}, {1,3}, {2,4,5}, {1,3,5} and {3,4} - all different.

R. C. Read, The Enumeration of Mating-Type Graphs. Report CORR 89-38, Dept. Combinatorics and Optimization, Univ. Waterloo, 1989.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrew Howroyd, Table of n, a(n) for n = 0..50
I. M. Gessel and J. Li, Enumeration of Point-Determining Graphs, arXiv:0705.0042 [math.CO], 2007-2009.
I. M. Gessel and J. Li, Enumeration of point-determining graphs, J. Combinatorial Theory Ser. A 118 (2011), 591-612.
R. C. Read, The Enumeration of Mating-Type Graphs, Dept. Combinatorics and Optimization, Univ. Waterloo, Oct 1989. (Annotated scanned copy)
D. Sumner, Point determination in graphs, Discrete Mathematics 5 (1973), 179-187.

Cf. A006025.
Cf. bi-point-determining graphs: labeled A129583, unlabeled A129584; connected bi-point-determining graphs: labeled A129585, unlabeled A129586; phylogenetic trees: labeled A000311, unlabeled A000669.
Cf. A007833, A079306 (connected)

a[n_] := Sum[StirlingS1[n, k] 2^Binomial[k, 2], {k, 0, n}];
Array[a, 15] (* Jean-François Alcover, Jul 25 2018 *)

a(n)=n!*polcoeff(sum(k=0,n,2^(k*(k-1)/2)*log(1+x+x*O(x^n))^k/k!),n) \\ Paul D. Hanna, May 20 2009

a(n) = Sum_{k=0..n} Stirling1(n, k)*2^binomial(k, 2). - Ronald C. Read and Vladeta Jovovic, Feb 10 2003

E.g.f.: Sum_{n>=0} 2^(n(n-1)/2)*log(1+x)^n/n!. - Paul D. Hanna, May 20 2009

More terms from Ronald C. Read and Vladeta Jovovic, Feb 10 2003

a(0)=1 prepended by Andrew Howroyd, Sep 09 2018

A006351 Number of series-parallel networks with n labeled edges. Also called yoke-chains by Cayley and MacMahon.

Original entry on oeis.org

1, 2, 8, 52, 472, 5504, 78416, 1320064, 25637824, 564275648, 13879795712, 377332365568, 11234698041088, 363581406419456, 12707452084972544, 477027941930515456, 19142041172838025216, 817675811320888020992, 37044610820729973813248, 1774189422608238694776832

Offset: 1

N. J. A. Sloane

For a simple relationship to series-reduced rooted trees, partitions of n, and phylogenetic trees among other combinatoric constructs, see comments in A000311. - Tom Copeland, Jan 06 2021

			D^3(1) = (12*x^2+56*x+52)/(x-1)^6. Evaluated at x = 0 this gives a(4) = 52.
a(3) = 8: The 8 possible increasing plane trees on 3 vertices with vertices of outdegree k >= 1 coming in 2 colors, B or W, are
.......................................................
.1B..1B..1W..1W.....1B.......1W........1B........1W....
.|...|...|...|...../.\....../..\....../..\....../..\...
.2B..2W..2B..2W...2...3....2....3....3....2....3....2..
.|...|...|...|.........................................
.3...3...3...3.........................................
G.f. = x + 2*x^2 + 8*x^3 + 52*x^4 + 472*x^5 + 5504*x^6 + 78416*x^7 + ...

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, p. 417.
P. A. MacMahon, Yoke-trains and multipartite compositions in connexion with the analytical forms called "trees", Proc. London Math. Soc. 22 (1891), 330-346; reprinted in Coll. Papers I, pp. 600-616. Page 333 gives A000084 = 2*A000669.
P. A. MacMahon, The combination of resistances, The Electrician, 28 (1892), 601-602; reprinted in Coll. Papers I, pp. 617-619.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 142.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.40(a), S(x).

Alois P. Heinz, Table of n, a(n) for n = 1..200
F. Bergeron, Ph. Flajolet and B. Salvy, Varieties of Increasing Trees, Lecture Notes in Computer Science vol. 581, ed. J.-C. Raoult, Springer 1992, pp. 24-48.
Veronica Bitonti, Bishal Deb, and Alan D. Sokal, Thron-type continued fractions (T-fractions) for some classes of increasing trees, arXiv:2412.10214 [math.CO], 2024. See p. 58.
Peter J. Cameron, Counting two-graphs related to trees, Elec. J. Combin., Vol. 2, #R4. [From _Aaron Meyerowitz_, Sep 30 2010]
Frédéric Chapoton, Florent Hivert, and Jean-Christophe Novelli, A set-operad of formal fractions and dendriform-like sub-operads, arXiv preprint arXiv:1307.0092 [math.CO], 2013
Diego Dominici, Nested derivatives: A simple method for computing series expansions of inverse functions, arXiv:math/0501052 [math.CA], 2005.
Brian Drake, An inversion theorem for labeled trees and some limits of areas under lattice paths (Example 1.5.1 ), Ph. D. dissertation, Brandeis Univ., 2008.
Steven R. Finch, Series-parallel networks
Steven R. Finch, Series-parallel networks, July 7, 2003. [Cached copy, with permission of the author]
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018.
ISCGI, Cograph graphs
Song He, Fei Teng, and Yong Zhang, String Correlators: Recursive Expansion, Integration-by-Parts and Scattering Equations, arXiv:1907.06041 [hep-th], 2019. Also in Journal of High Energy Physics (2019), 2019:85.
Walter Knödel, Über Zerfällungen, Monatsh. Math., 55 (1951), 20-27.
Vladimir Kruchinin, The method for obtaining expressions for coefficients of reverse generating functions, arXiv:1211.3244 [math.CO], 2012.
Z. A. Lomnicki, Two-terminal series-parallel networks, Adv. Appl. Prob., 4 (1972), 109-150.
Michael D. Moffitt, Search Strategies for Topological Network Optimization, Proceedings of the AAAI Conference on Artificial Intelligence, 36(9) (2022), 10299-10308.
John W. Moon, Some enumerative results on series-parallel networks, Annals Discrete Math., 33 (1987), 199-226 (the e.g.f. U(x)).
N. J. A. Sloane, Transforms
Index entries for reversions of series
Index entries for sequences mentioned in Moon (1987)
Index entries for sequences related to trees [From _Aaron Meyerowitz_, Sep 30 2010]

Cf. A000311, A000084 (for unlabeled case), A032188. A140945.

Cf. A133314, A134991, A135494.

read transforms; t1 := 2*ln(1+x)-x; t2 := series(t1,x,10); t3 := seriestoseries(t2,'revogf'); t4 := SERIESTOLISTMULT(%);
# N denotes all series-parallel networks, S = series networks, P = parallel networks;
spec := [ N, N=Union(Z,S,P),S=Set(Union(Z,P),card>=2), P=Set(Union(Z,S), card>=2)}, labeled ]: A006351 := n->combstruct[count](spec,size=n);
A006351 := n -> add(combinat[eulerian2](n-1,k)*2^(n-k-1),k=0..n-1):
seq(A006351(n), n=1..18); # Peter Luschny, Nov 16 2012

max = 18; f[x_] := 2*Log[1+x]-x; Rest[ CoefficientList[ InverseSeries[ Series[ f[x], {x, 0, max}], x], x]]*Range[max]! (* Jean-François Alcover, Nov 25 2011 *)

a(n):=if n=1 then 1 else ((n-1)!*sum(binomial(n+k-1,n-1)* sum((-1)^(j)*binomial(k,j)*sum((binomial(j,l)*(j-l)!*2^(j-l)*(-1)^l* stirling1(n-l+j-1,j-l))/(n-l+j-1)!,l,0,j),j,1,k),k,1,n-1)); /* Vladimir Kruchinin, Jan 24 2012 */

x='x+O('x^66); Vec(serlaplace(serreverse( 2*log(1+x) - 1*x ))) \\ Joerg Arndt, May 01 2013

# uses[eulerian2 from A201637]
def A006351(n): return add(A201637(n-1, k)*2^(n-k-1) for k in (0..n-1))
[A006351(n) for n in (1..18)]  # Peter Luschny, Nov 16 2012

For n >= 2, A006351(n) = 2*A000311(n) = A005640(n)/2^n. Row sums of A140945.
E.g.f. is reversion of 2*log(1+x)-x.
Also exponential transform of A000311, define b by 1+sum b_n x^n / n! = exp ( 1 + sum a_n x^n /n!).
E.g.f.: A(x), B(x)=x*A(x) satisfies the differential equation B'(x)=(1+B(x))/(1-B(x)). - Vladimir Kruchinin, Jan 18 2011
From Peter Bala, Sep 05 2011: (Start)
The generating function A(x) satisfies the autonomous differential equation A'(x) = (1+A)/(1-A) with A(0) = 0. Hence the inverse function A^-1(x) = int {t = 0..x} (1-t)/(1+t) = 2*log(1+x)-x, which yields A(x) = -1-2*W(-1/2*exp((x-1)/2)), where W is the Lambert W function.
The expansion of A(x) can be found by inverting the above integral using the method of [Dominici, Theorem 4.1] to arrive at the result a(n) = D^(n-1)(1) evaluated at x = 0, where D denotes the operator g(x) -> d/dx((1+x)/(1-x)*g(x)). Compare with A032188.
Applying [Bergeron et al., Theorem 1] to the result x = int {t = 0..A(x)} 1/phi(t), where phi(t) = (1+t)/(1-t) = 1 + 2*t + 2*t^2 + 2*t^3 + ..., leads to the following combinatorial interpretation for the sequence: a(n) gives the number of plane increasing trees on n vertices where each vertex of outdegree k >=1 can be in one of 2 colors. An example is given below. (End)
A134991 gives (b.+c.)^n = 0^n , for (b_n)=A000311(n+1) and (c_0)=1, (c_1)=-1, and (c_n)=-2* A000311(n) = -A006351(n) otherwise. E.g., umbrally, (b.+c.)^2 = b_2*c_0 + 2 b_1*c_1 + b_0*c_2 =0. - Tom Copeland, Oct 19 2011
G.f.: 1/S(0) where S(k) = 1 - x*(k+1) - x*(k+1)/S(k+1); (continued fraction). - Sergei N. Gladkovskii, Dec 18 2011
a(n) = ((n-1)!*sum(k=1..n-1, C(n+k-1,n-1)*sum(j=1..k, (-1)^(j)*C(k,j)* sum(l=0..j, (C(j,l)*(j-l)!*2^(j-l)*(-1)^l*stirling1(n-l+j-1,j-l))/ (n-l+j-1)!)))), n>1, a(1)=1. - Vladimir Kruchinin, Jan 24 2012
E.g.f.: A(x) = exp(B(x))-1 where B(x) is the e.g.f. of A000311. - Vladimir Kruchinin, Sep 25 2012
a(n) = sum_{k=0..n-1} A201637(n-1,k)*2^(n-k-1). - Peter Luschny, Nov 16 2012
G.f.: -1 + 2/Q(0), where Q(k)= 1 - k*x - x*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 01 2013
a(n) ~ sqrt(2)*n^(n-1)/((2*log(2)-1)^(n-1/2)*exp(n)). - Vaclav Kotesovec, Jul 17 2013
G.f.: Q(0)/(1-x), where Q(k) = 1 - x*(k+1)/( x*(k+1) - (1 -x*(k+1))*(1 -x*(k+2))/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 10 2013
a(1) = 1; a(n) = a(n-1) + Sum_{k=1..n-1} binomial(n-1,k) * a(k) * a(n-k). - Ilya Gutkovskiy, Aug 28 2020
Conjecture: a(n) = A379459(n-2,0) = A379460(n-1,0) for n > 1 with a(1) = 1. - Mikhail Kurkov, Jan 16 2025

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

A134685 Irregular triangle read by rows: coefficients C(j,k) of a partition transform for direct Lagrange inversion.

Original entry on oeis.org

1, -1, 3, -1, -15, 10, -1, 105, -105, 10, 15, -1, -945, 1260, -280, -210, 35, 21, -1, 10395, -17325, 6300, 3150, -280, -1260, -378, 35, 56, 28, -1, -135135, 270270, -138600, -51975, 15400, 34650, 6930, -2100, -1575, -2520, -630, 126, 84, 36, -1

Offset: 1

Tom Copeland, Jan 26 2008, Sep 13 2008

Let f(t) = u(t) - u(0) = Ev[exp(u.* t) - u(0)] = log{Ev[(exp(z.* t))/z_0]} = Ev[-log(1- a.* t)], where the operator Ev denotes umbral evaluation of the umbral variables u., z. or a., e.g., Ev[a.^n + a.^m] = a_n + a_m . The relation between z_n and u_n is given in reference in A127671 and u_n = (n-1)! * a_n .
If u_1 is not equal to 0, then the compositional inverse for these expressions is given by g(t) = Sum_{j>=1} P(j,t) where, with u_n denoted by (n') for brevity,
P(1,t) = (1')^(-1) * [ 1 ] * t
P(2,t) = (1')^(-3) * [ -(2') ] * t^2 / 2!
P(3,t) = (1')^(-5) * [ 3 (2')^2 - (1')(3') ] * t^3 / 3!
P(4,t) = (1')^(-7) * [ -15 (2')^3 + 10 (1')(2')(3') - (1')^2 (4') ] * t^4 / 4!
P(5,t) = (1')^(-9) * [ 105 (2')^4 - 105 (1') (2')^2 (3') + 10 (1')^2 (3')^2 + 15 (1')^2 (2') (4') - (1')^3 (5') ] * t^5 / 5!
P(6,t) = (1')^(-11) * [ -945 (2')^5 + 1260 (1') (2')^3 (3') - 280 (1')^2 (2') (3')^2 - 210 (1')^2 (2')^2 (4') + 35 (1')^3 (3')(4') + 21 (1')^3 (2')(5') - (1')^4 (6') ] * t^6 / 6!
P(7,t) = (1')^(-13) * [ 10395 (2')^6 - 17325 (1') (2')^4 (3') +  (1')^2 [ 6300 (2')^2 (3')^2 + 3150  (2')^3 (4')] - (1')^3 [280 (3')^3 + 1260  (2')(3')(4') + 378  (2')^2(5')] + (1')^4 [35 (4')^2 + 56 (3')(5') + 28 (2')(6')]  - (1')^5 (7') ] * t^7 / 7!
P(8,t) = (1')^(-15) * [ -135135 (2')^7 + 270270 (1') (2')^5 (3') -  (1')^2 [ 138600 (2')^3 (3')^2 +  51975 (2')^4 (4')] + (1')^3 [15400 (2')(3')^3 + 34650  (2')^2(3')(4') + 6930  (2')^3(5')] - (1')^4 [2100 (3')^2(4') + 1575 (2')(4')^2 + 2520 (2')(3')(5') + 630 (2')^2(6') ]  + (1')^5 [126 (4')(5') + 84 (3')(6') + 36 (2')(7')] - (1')^6 (8') ] * t^8 / 8!
...
Substituting ((m-1)') for (m') in each partition and ignoring the (0') factors, the partitions in the brackets of P(n,t) become those of n-1 listed in Abramowitz and Stegun on page 831 (in the reversed order) and the number of partitions in P(n,t) is given by A000041(n-1).
Combinatorial interpretations are given in the link.
From Tom Copeland, Jul 10 2018: (Start)
Coefficients occurring in prolongation for the special Euclidean group SE(2) and special affine group SA(2) in the Olver presentation on moving frames (MFP) in slides 33 and 42. These are a result of applying an iterated derivative of the form h(x)d/dx = d/dy as in this entry (more generally as g(x) d/dx as discussed in A145271). See also p. 6 of Olver's paper on contact forms, but note that the 12 should be a 15 in the formula for the compositional inverse of S(t).
Change variables in the MFP to obtain connections to the partition polynomials Prt_n = n! * P(n,1) above. Let delta and beta in the formulas for the equi-affine curves in MFP be L and B, respectively, and D_y = (1/(L-B*u_x)) d/dx = (1/w_x) d/dx. Then v_(yy) = (1/B) [-w_(xx)/(w_x)^3] in MFP (there is an overall sign error in MFP for v_(yy) and higher derivatives w.r.t. y), and (d/dy)^n v = v_n = (1/B)* [(1/w_1)*(d/dx)]^(n-2) [-w_2/(w_1)^3] for n > 1, with w_n = (d/dx)^n w. Consequently, in the partition polynomials Prt_n for n > 1 here substitute (n') = -B*u_n = w_n for n > 1 and (1')  = L-B*u_1 = w_1, where u_n = (d/dx)^n u, and then divide by B. For example, v_4 = (1/B)*Prt_4 = (1/B)*4!*P(4,1) = (1/B) (L-B*u_n)^(-7) [-15*(-B*u_2)^3 + 10 (L-B*u_1)(-B*u_2)(-B*u_3) - (L-B*u_1)^2 (-B*u_4)], agreeing with v_4 in MFP except for the overall sign.
For the SE(2) transformation formulas in MFP, let w_x = cos(phi) + sin(phi)*u_x, and then the same transformations apply as above with cos(phi) and sin(phi) substituted for L and -B, respectively. (End)

			Examples and checks:
1) Let u_1 = -1 and u_n = 1 for n>1,
then f(t) = exp(u.*t) - u(0) = exp(t)-2t-1
and g(t) = [e.g.f. of signed A000311];
therefore, the row sums of unsigned [C(j,k)] are A000311 =
(0,1,1,4,26,236,2752,...) = (0,-P(1,1),2!*P(2,1),-3!*P(3,1),4!*P(4,1),...).
2) Let u_1 = -1 and u_n = (n-1)! for n>1,
then f(t) = -log(1-t)-2t
and g(t) = [e.g.f. of signed (0,A032188)]
with (0,A032188) = (0,1,1,5,41,469,6889,...) = (0,-P(1,1),2!*P(2,1),-3!P(3,1),...).
3) Let u_1 = -1 and u_n = (-1)^n (n-2)! for n>1, then
f(t) = (1+t) log(1+t) - 2t
and g(t) = [e.g.f. of signed (0,A074059)]
with (0,A074059) = (0,1,1,2,7,34,213,...) = (0,-P(1,1),2!*P(2,1),-3!*P(3,1),...).
4) Let u_1 = 1, u_2 = -1 and u_n = 0 for n>2,
then f(t) = t(1-t/2)
and g(t) = [e.g.f. of (0,A001147)] = 1 - (1-2t)^(1/2)
with (0,A001147) = (0,1,1,3,15,105,945...) =(0,P(1,1),2!*P(2,1),3!*P(3,1),...).
5) Let u_1 = 1, u_2 = -2 and u_n = 0 for n>2,
then f(t)= t(1-t)
and g(t) = t * [o.g.f. of A000108] = [1 - (1-4t)^(1/2)] / 2
with (0,A000108) = (0,1,1,2,5,14,42,...) = (0,P(1,1),P(2,1),P(3,1),...).
.
From _Peter Luschny_, Feb 19 2021: (Start)
Triangle starts:
 [1]  1;
 [2] -1;
 [3]  3,     -1;
 [4] -15,     10,    -1;
 [5]  105,   -105,   [10, 15],  -1;
 [6] -945,    1260,  [-280, -210], [35, 21],  -1;
 [7]  10395, -17325, [6300, 3150], [-280, -1260, -378], [35, 56, 28], -1;
 [8] -135135, 270270, [-138600, -51975], [15400, 34650, 6930], [-2100, -1575, -2520, -630], [126, 84, 36], -1
The coefficients can be seen as a refinement of the Ward numbers: Let R(n, k) = Sum T(n, k), where the sum collects adjacent terms with equal sign, as indicated by the square brackets in the table, then R(n+1, k+1) = (-1)^(n-k)*W(n, k), where W(n, k) are the Ward numbers A181996, for n >= 0 and 0 <= k <= n-1.  (End)

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 831.
D. S. Alexander, A History of Complex Dynamics: From Schröder to Fatou to Julia, Friedrich Vieweg & Sohn, 1994, p. 10.
J. Riordan, Combinatorial Identities, Robert E. Krieger Pub. Co., 1979, (unsigned partition polynomials in Table 5.2 on p. 181, but may have errors).

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
O. Arnaldsson, Élie Cartan’s Theory of Moving Frames, slide presentation, 2014.
Tom Copeland, Generators, Inversion, and Matrix, Binomial, and Integral Transforms, 2015.
Tom Copeland, Important formulas in combinatorics, MathOverflow answer, 2015.
Tom Copeland, Compositional inversion and generating functions in algebraic geometry, MathOverflow question, 2014.
Tom Copeland, Compositional inverse pairs, the inviscid Burgers-Hopf equation, and the Stasheff associahedra, 2014.
Tom Copeland, Lagrange a la Lah, 2011.
Tom Copeland, Short Note on Lagrange Inversion, 2008.
Tom Copeland, Formal group laws and binomial Sheffer sequences, 2018.
Bishal Deb and Alan D. Sokal, Higher-order Stirling cycle and subset triangles: Total positivity, continued fractions and real-rootedness, arXiv:2507.18959 [math.CO], 2025. See p. 32.
Hector Figueroa and Jose M. Gracia-Bondia, Combinatorial Hopf algebras in quantum field theory I, arXiv:0408145 [hep-th], 2005, (p. 38).
Ezra Getzler, The semi-classical approximation for modular operads, arXiv:alg-geom/9612005, 1996 (see p. 2).
Peter J. Olver, The canonical contact form.
Peter J. Olver, Moving Frames, slide presentation, 2009.
Andrew Elvey Price and Alan D. Sokal, Phylogenetic trees, augmented perfect matchings, and a Thron-type continued fraction (T-fraction) for the Ward polynomials, arXiv:2001.01468 [math.CO], 2020.
Jair Patrick Taylor, Formal group laws and hypergraph colorings, doctoral thesis, Univ. of Wash., 2016, p. 95.

Cf. A145271, (A134991, A019538) = (reduced array, associated g(x)).
Cf. A000108, A000311, A001147, A032188, A049019, A074059, A129185, A133314, A139605, A036040.
Cf. A181996 (Ward numbers).

rows[n_] := {{1}}~Join~Module[{h = 1/(1 + Sum[u[k] y^k/k!, {k, n-1}] + O[y]^n), g = y, r}, r = Reap[Do[g = h D[g, y]; Sow[Expand[Normal@g /. {y -> 0}]], {k, n}]][[2, 1, ;;]]; Table[Coefficient[r[[k]], Product[u[t], {t, p}]], {k, 2, n}, {p, Reverse@Sort[Sort /@ IntegerPartitions[k-1]]}]];
rows[8] // Flatten (* Andrei Zabolotskii, Feb 19 2024 *)

The bracketed partitions of P(n,t) are of the form (u_1)^e(1) (u_2)^e(2) ... (u_n)^e(n) with coefficients given by (-1)^(n-1+e(1)) * [2*(n-1)-e(1)]! / [2!^e(2)*e(2)!*3!^e(3)*e(3)! ... n!^e(n)*e(n)! ].
From Tom Copeland, Sep 05 2011: (Start)
Let h(t) = 1/(df(t)/dt)
  = 1/Ev[u.*exp(u.*t)]
  = 1/(u_1+(u_2)*t+(u_3)*t^2/2!+(u_4)*t^3/3!+...),
  an e.g.f. for the partition polynomials of A133314
  (signed A049019) with an index shift.
  Then for the partition polynomials of A134685,
  n!*P(n,t) = ((t*h(y)*d/dy)^n) y evaluated at y=0,
  and the compositional inverse of f(t) is
  g(t) = exp(t*h(y)*d/dy) y evaluated at y=0.
  Also, dg(t)/dt = h(g(t)). (Cf. A000311 and A134991)(End)
From Tom Copeland, Oct 30 2011: (Start)
With exp[x* PS(.,t)] = exp[t*g(x)]=exp[x*h(y)d/dy] exp(t*y) eval. at y=0, the raising/creation and lowering/annihilation operators
defined by R PS(n,t)=PS(n+1,t) and L PS(n,t)= n*PS(n-1,t) are
R = t*h(d/dt) = t * 1/[u_1+(u_2)*d/dt+(u_3)*(d/dt)^2/2!+...],  and
L = f(d/dt)=(u_1)*d/dt+(u_2)*(d/dt)^2/2!+(u_3)*(d/dt)^3/3!+....
Then  P(n,t) = (t^n/n!) dPS(n,z)/dz  eval. at z=0. (Cf. A139605, A145271, and link therein to Mathemagical Forests for relation to planted trees on p. 13.) (End)
The bracketed partition polynomials of P(n,t) are also given by (d/dx)^(n-1) 1/[u_1 + u_2 * x/2! + u_3 * x^2/3! + ... + u_n * x^(n-1)/n!]^n evaluated at x=0. - Tom Copeland, Jul 07 2015
Equivalent matrix computation: Multiply the m-th diagonal (with m=1 the index of the main diagonal) of the lower triangular Pascal matrix by u_m = (d/dx)^m f(x) evaluated at x=0 to obtain the matrix UP with UP(n,k) = binomial(n,k) u_{n+1-k}. Then P(n,t) = (1, 0, 0, 0, ...) [UP^(-1) * S]^(n-1) FC * t^n/n!, where S is the shift matrix A129185, representing differentiation in the basis x^n//n!, and FC is the first column of UP^(-1), the inverse matrix of UP. These results follow from A145271 and A133314. - Tom Copeland, Jul 15 2016
Also, P(n,t) = (1, 0, 0, 0, ...) [UP^(-1) * S]^n (0, 1, 0, ..)^T * t^n/n! in agreement with A139605. - Tom Copeland, Aug 27 2016
From Tom Copeland, Sep 20 2016: (Start)
Let PS(n,u1,u2,...,un) = P(n,t) / (t^n/n!), i.e., the square-bracketed part of the partition polynomials in the expansion for the inverse in the comment section, with u_k = uk.
Also let PS(n,u1=1,u2,...,un) = PB(n,b1,b2,...,bK,...) where each bK represents the partitions of PS, with u1 = 1, that have K components or blocks, e.g., PS(5,1,u2,...,u5) = PB(5,b1,b2,b3,b4) = b1 + b2 + b3 + b4 with b1 = -u5, b2 = 15 u2 u4 + 10 u3^2, b3 = -105 u2^2 u3,  and b4 = 105 u2^4.
The relation between solutions of the inviscid Burgers' equation and compositional inverse pairs (cf. link and A086810) implies, for n > 2,  PB(n, 0 * b1, 1 * b2,..., (K-1) * bK, ...) = (1/2) * Sum_{k = 2..n-1} binomial(n+1,k) * PS(n-k+1,u_1=1,u_2,...,u_(n-k+1)) * PS(k,u_1=1,u_2,...,u_k).
For example, PB(5,0 * b1, 1 * b2, 2 * b3, 3 * b4) = 3 * 105 u2^4 - 2 * 105 u2^2 u3 + 1 * 15 u2 u4 + 1 * 10 u3^2 - 0 * u5 = 315 u2^4 - 210 u2^2 u3 + 15 u2 u4 + 10 u3^2 = (1/2) [2 * 6!/(4!*2!) * PS(2,1,u2) * PS(4,1,u2,...,u4) + 6!/(3!*3!) * PS(3,1,u2,u3)^2] = (1/2) * [ 2 * 6!/(4!*2!)  * (-u2) (-15 u2^3 + 10 u2 u3 - u4) + 6!/(3!*3!) * (3 u2^2 - u3)^2].
Also, PB(n,0*b1,1*b2,...,(K-1)*bK,...) =  d/dt t^(n-2)*PS(n,u1=1/t,u2,...,un)|{t=1} = d/dt (1/t)*PS(n,u1=1,t*u2,...,t*un)|{t=1}.
(End)
A recursion relation for computing each partition polynomial of this entry from the lower order polynomials and the coefficients of the Bell polynomials of A036040 is presented in the blog entry "Formal group laws and binomial Sheffer sequences." - Tom Copeland, Feb 06 2018

P(7,t) and P(8,t) data added by Tom Copeland, Jan 14 2016

Terms in rows 5-8 reordered by Andrei Zabolotskii, Feb 19 2024

cp's OEIS Frontend

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