A000169 -id:A000169 - cp's OEIS frontend

A000055 Number of trees with n unlabeled nodes.

1, 1, 1, 1, 2, 3, 6, 11, 23, 47, 106, 235, 551, 1301, 3159, 7741, 19320, 48629, 123867, 317955, 823065, 2144505, 5623756, 14828074, 39299897, 104636890, 279793450, 751065460, 2023443032, 5469566585, 14830871802, 40330829030, 109972410221, 300628862480, 823779631721, 2262366343746, 6226306037178

Offset: 0

N. J. A. Sloane

Also, number of unlabeled 2-gonal 2-trees with n-1 2-gons, for n>0. [Corrected by Andrei Zabolotskii, Jul 29 2025]
Main diagonal of A054924.
Left border of A157905. - Gary W. Adamson, Mar 08 2009
From Robert Munafo, Jan 24 2010: (Start)
Also counts classifications of n items that require exactly n-1 binary partitions; see Munafo link at A005646, also A171871 and A171872.
The 11 trees for n = 7 are illustrated at the Munafo web link.
Link to A171871/A171872 conjectured by Robert Munafo, then proved by Andrew Weimholt and Franklin T. Adams-Watters on Dec 29 2009. (End)
This is also "Number of tree perfect graphs on n nodes" [see Hougardy]. - N. J. A. Sloane, Dec 04 2015
For n > 0, a(n) is the number of ways to arrange n-1 unlabeled non-intersecting circles on a sphere. - Vladimir Reshetnikov, Aug 25 2016
All trees for n=1 through n=12 are depicted in Chapter 1 of the Steinbach reference. On p. 6 appear encircled two trees (with n=10) which seem inequivalent only when considered as ordered (planar) trees. Earlier instances of such possibly (in)equivalent trees could appear from n=6 on (and from n=9 on without equivalence modulo plane symmetry) but are not drawn separately there. - M. F. Hasler, Aug 29 2017

			a(1) = 1 [o]; a(2) = 1 [o-o]; a(3) = 1 [o-o-o];
a(4) = 2 [o-o-o and o-o-o-o]
            |
            o
G.f. = 1 + x + x^2 + x^3 + 2*x^4 + 3*x^5 + 6*x^6 + 11*x^7 + 23*x^8 + ...

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 279.
Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 55.
N. L. Biggs et al., Graph Theory 1736-1936, Oxford, 1976, p. 49.
A. Cayley, On the analytical forms called trees, with application to the theory of chemical combinations, Reports British Assoc. Advance. Sci. 45 (1875), 257-305 = Math. Papers, Vol. 9, 427-460 (see p. 459).
S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 295-316.
J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 526.
F. Harary, Graph Theory. Addison-Wesley, Reading, MA, 1969, p. 232.
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 58 and 244.
D. E. Knuth, Fundamental Algorithms, 3d Ed. 1997, pp. 386-88.
R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford, 1998.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 138.
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).

R. J. Mathar and Robert G. Wilson v, Table of n, a(n) for n = 0..1000
Jeremy C. Adcock, Sam Morley-Short, Joshua W. Silverstone, and Mark G. Thompson, Hard limits on the postselectability of optical graph states, arXiv:1806.03263 [quant-ph], 2018.
H. R. Afshar, E. A. Bergshoeff, and W. Merbis, Interacting spin-2 fields in three dimensions, arXiv preprint arXiv:1410.6164 [hep-th], 2014-2015.
Ruggero Bandiera and Florian Schaetz, Eulerian idempotent, pre-Lie logarithm and combinatorics of trees, arXiv:1702.08907 [math.CO], 2017. Mentions this sequence.
Madeleine Burkhart and Joel Foisy, Enumerating spherical n-links, Involve, Vol. 11 (2018), No. 2, 195-206.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
A. Cayley, On the analytical forms called trees, Amer. J. Math., 4 (1881), 266-268.
CombOS - Combinatorial Object Server, Generate graphs
Benjamin Côté, Hélène Cossette, and Etienne Marceau, Centrality and topology properties in a tree-structured Markov random field, arXiv:2410.20240 [math.ST], 2024. See p. 18.
Antoine Dailly and Tuomo Lehtilä, Reconstructing graphs with subgraph compositions, arXiv:2504.00169 [math.CO], 2025. See p. 28.
R. Ferrer-i-Cancho, Non-crossing dependencies: least effort, not grammar, arXiv preprint arXiv:1411.2645 [cs.CL], 2014.
S. R. Finch, Otter's Tree Enumeration Constants
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 481.
Andrew Gainer-Dewar, Gamma-Species and the Enumeration of k-Trees, Electronic Journal of Combinatorics, Volume 19 (2012), #P45 and arXiv:1208.5993 [math.CO], 2012.
Xiangrui Gao, Song He, and Yong Zhang, Labelled tree graphs, Feynman diagrams and disk integrals arxiv:1708.08701 [hep-th], 2017, see p. 4.
Ira M. Gessel, Good Will Hunting's Problem: Counting Homeomorphically Irreducible Trees, arXiv:2305.03157 [math.CO], 2023.
D. D. Grant, The stability index of graphs, pp. 29-52 of Combinatorial Mathematics (Proceedings 2nd Australian Conf.), Lect. Notes Math. 403, 1974. Gives first 45 terms.
Paul E. Gunnells, Generalized Catalan numbers from hypergraphs, arXiv:2102.05121 [math.CO], 2021. Mentions this sequence p. 3.
T. Hoppe and A. Petrone, Integer sequence discovery from small graphs, arXiv preprint arXiv:1408.3644 [math.CO], 2014.
S. Hougardy, Classes of perfect graphs, Discr. Math. 306 (2006), 2529-2571.
House of Graphs, Trees
Andrew Jobbings, Enumerating nets, Preprint 2015.
Mithra Karamchedu and Lucas Bang, Generating the Spanning Trees of Series-Parallel Graphs up to Graph Automorphism, arXiv:2508.13480 [cs.DS], 2025. See p. 4.
Elena V. Konstantinova and Maxim V. Vidyuk, Discriminating tests of information and topological indices. Animals and trees, J. Chem. Inf. Comput. Sci. 43 (2003), 1860-1871. See Table 15, column 1 on page 1868.
G. Labelle, C. Lamathe and P. Leroux, Labeled and unlabeled enumeration of k-gonal 2-trees, arXiv:math/0312424 [math.CO], 2003.
Steve Lawford and Yll Mehmeti, Cliques and a new measure of clustering: with application to U.S. domestic airlines, arXiv:1806.05866 [cs.SI], 2018.
P. Leroux and B. Miloudi, Généralisations de la formule d'Otter, Ann. Sci. Math. Québec, Vol. 16, No. 1, pp. 53-80, 1992. (Annotated scanned copy)
Gang Li, Generation of Rooted Trees and Free Trees, Comp. Sci. MSc thesis paper, 1996.
R. J. Mathar, Topologically Distinct Sets of Non-intersecting Circles in the Plane, arXiv:1603.00077 [math.CO], 2016.
Richard J. Mathar, Counting Connected Graphs without Overlapping Cycles, arXiv:1808.06264 [math.CO], 2018.
Robert Munafo, Relation to Tree Graphs
R. Otter, The number of trees, Ann. of Math. (2) 49 (1948), 583-599.
Igor Pak, Complexity problems in enumerative combinatorics, arXiv:1803.06636 [math.CO], 2018.
E. M. Palmer and A. J. Schwenk, On the number of trees in a random forest, J. Combin. Theory, B 27 (1979), 109-121.
Sebastian Piec, Krzysztof Malarz and Krzysztof Kulakowski. How to count trees?, arXiv:cond-mat/0501594 [cond-mat.stat-mech], 2005; Int. J. Modern Phys., C16 (2005) 1527-1534.
N. Pippenger, Enumeration of equicolorable trees, SIAM J. Discrete Math., 14 (2001), 93-115.
J. M. Plotkin and J. W. Rosenthal, How to obtain an asymptotic expansion of a sequence from an analytic identity satisfied by its generating function, J. Austral. Math. Soc. (Series A), Vol. 56 (1994), 131-143.
E. M. Rains and N. J. A. Sloane, On Cayley's Enumeration of Alkanes (or 4-Valent Trees), J. Integer Sequences, Vol. 2 (1999), Article 99.1.1.
R. W. Robinson, Letter to N. J. A. Sloane, Jul 29 1980
Yaniv Sadeh, Haim Kaplan, and Uri Zwick, Search Trees on Trees via LP, arXiv:2501.17563 [cs.DS], 2025. See p. 12.
Sage, Common Graphs
A. J. Schwenk, Letter to N. J. A. Sloane, Aug 1972
N. J. A. Sloane, Illustration of initial terms
Peter Steinbach, Field Guide to Simple Graphs, Volume 3, Overview of the following parts: Cover, Front matter, Chapter 1: Trees, Trees (cont'd: pt.2), Trees (cont'd: pt.3), Chapter 2: Centers and Centroids, Chap.2 (cont'd), Chapter 3: Random Trees, Chapter 4: Rooted Trees, Chapter 5: Homeomorphically Irreducible Trees, Chapter 6: Tables (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
Ivan Stošić and Ivan Damnjanović, An efficient algorithm for generating transmission irregular trees, arXiv:2502.15453 [cs.DM], 2025. See p. 11.
Tanay Wakhare, Eric Wityk, and Charles R. Johnson. The proportion of trees that are linear, Discrete Mathematics 343.10 (2020): 112008. Also arXiv:1901.08502 [math.CO], 2019-2020. See Tables 1 and 2 (but beware errors).
Eric Weisstein's World of Mathematics, Tree
Pascal Welke, Tamás Horváth, and Stefan Wrobel, Probabilistic and exact frequent subtree mining in graphs beyond forests, Machine Learning (2019), 1-28.
Robert Alan Wright, Bruce Richmond, Andrew Odlyzko, and Brendan D. McKay, Constant Time Generation of Free Trees, SIAM Journal of Computing, vol. 15, no. 2, pp. 540-548, (1986) [preprint].
Index entries for sequences related to trees
Index entries for "core" sequences

Cf. A000676 (centered trees), A000677 (bicentered trees), A027416 (trees with a centroid), A102011 (trees with a bicentroid), A034853 (refined by diameter), A238414 (refined by maximum vertex degree).
Cf. A000081 (rooted trees), A000272 (labeled trees), A000169 (labeled rooted trees), A212809 (radius of convergence).
Cf. A036361 (labeled 2-trees), A036362 (labeled 3-trees), A036506 (labeled 4-trees), A054581 (unlabeled 2-trees).
Cf. A157904, A157905, A005195 (Euler transform = forests), A095133 (multisets).
Column 0 of A335362 and A034799.
Related to A005646; see A171871 and A171872.
Cf. also A000088, A008406, A051491, A086308.

import Data.List (generic_index)
import Math.OEIS (getSequenceByID)
triangle x = (x * x + x) `div` 2
a000055 n = let {r = genericIndex (fromJust (getSequenceByID "A000081")); (m, nEO) = divMod n 2}
            in  r n - sum (zipWith (*) (map r [0..m]) (map r [n, n-1..]))
                + (1-nEO) * (triangle (r m + 1))
-- Walt Rorie-Baety, Jun 12 2021

N := 30; P := PowerSeriesRing(Rationals(),N+1); f := func< A | x*&*[Exp(Evaluate(A,x^k)/k) : k in [1..N]]>; G := x; for i in [1..N] do G := f(G); end for; G000081 := G; G000055 := 1 + G - G^2/2 + Evaluate(G,x^2)/2; A000055 := Eltseq(G000055); // Geoff Baileu (geoff(AT)maths.usyd.edu.au), Nov 30 2009

G000055 := series(1+G000081-G000081^2/2+subs(x=x^2,G000081)/2,x,31); A000055 := n->coeff(G000055,x,n); # where G000081 is g.f. for A000081 starting with n=1 term
with(numtheory): b:= proc(n) option remember; `if`(n<=1, n, (add(add(d*b(d), d=divisors(j)) *b(n-j), j=1..n-1))/ (n-1)) end: a:= n-> `if`(n=0, 1, b(n) -(add(b(k) *b(n-k), k=0..n) -`if`(irem(n, 2)=0, b(n/2), 0))/2):
seq(a(n), n=0..50);
# Alois P. Heinz, Aug 21 2008
# Program to create b-file b000055.txt:
A000081 := proc(n) option remember; local d, j;
if n <= 1 then n else
    add(add(d*procname(d),d=numtheory[divisors](j))*procname(n-j),j=1..n-1)/(n-1);
fi end:
A000055 := proc(nmax) local a81, n, t, a, j, i ;
a81 := [seq(A000081(i), i=0..nmax)] ; a := [] ;
for n from 0 to nmax do
    if n = 0 then
        t := 1+op(n+1, a81) ;
    else
        t := op(n+1, a81) ;
    fi;
    if type(n, even) then
        t := t-op(1+n/2, a81)^2/2 ;
        t := t+op(1+n/2, a81)/2 ;
    fi;
    for j from 0 to (n-1)/2 do
        t := t-op(j+1, a81)*op(n-j+1, a81) ;
    od:
    a := [op(a), t] ;
od:
a end:
L := A000055(1000) ;
#  R. J. Mathar, Mar 06 2009

s[n_, k_] := s[n, k] = a[n + 1 - k] + If[n < 2k, 0, s[n - k, k]]; a[1] = 1; a[n_] := a[n] = Sum[a[i] s[n-1, i] i, {i, 1, n-1}] / (n-1); Table[a[i] - Sum[a[j] a[i-j], {j, 1, i/2}] + If[OddQ[i], 0, a[i/2] (a[i/2] + 1)/2], {i, 1, 50}] (* Robert A. Russell *)
b[0] = 0; b[1] = 1; b[n_] := b[n] = Sum[d*b[d]*b[n-j], {j, 1, n-1}, {d, Divisors[j]}]/(n-1); a[0] = 1; a[n_] := b[n] - (Sum[b[k]*b[n-k], {k, 0, n}] - If[Mod[n, 2] == 0, b[n/2], 0])/2; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Apr 09 2014, after Alois P. Heinz *)

{a(n) = local(A, A1, an, i, t); if( n<2, n>=0, an = Vec(A = A1 = 1 + O('x^n)); for(m=2, n, i=m\2; an[m] = sum(k=1, i, an[k] * an[m-k]) + (t = polcoeff( if( m%2, A *= (A1 - 'x^i)^-an[i], A), m-1))); t + if( n%2==0, binomial( -polcoeff(A, i-1), 2)))}; /* Michael Somos */

N=66;  A=vector(N+1, j, 1);
for (n=1, N, A[n+1] = 1/n * sum(k=1, n, sumdiv(k, d, d * A[d]) * A[n-k+1] ) );
A000081=concat([0], A);
H(t)=subst(Ser(A000081, 't), 't, t);
x='x+O('x^N);
Vec( 1 + H(x) - 1/2*( H(x)^2 - H(x^2) ) )
\\ Joerg Arndt, Jul 10 2014

# uses function from A000081
def A000055(n): return 1 if n == 0 else A000081(n)-sum(A000081(i)*A000081(n-i) for i in range(1,n//2+1)) + (0 if n % 2 else (A000081(n//2)+1)*A000081(n//2)//2) # Chai Wah Wu, Feb 03 2022

[len(list(graphs.trees(n))) for n in range(16)] # Peter Luschny, Mar 01 2020

G.f.: A(x) = 1 + T(x) - T^2(x)/2 + T(x^2)/2, where T(x) = x + x^2 + 2*x^3 + ... is the g.f. for A000081.
a(n) ~ A086308 * A051491^n * n^(-5/2). - Vaclav Kotesovec, Jan 04 2013
a(n) = A000081(n) - A217420(n+1), n > 0. - R. J. Mathar, Sep 19 2016
a(n) = A000676(n) + A000677(n). - R. J. Mathar, Aug 13 2018
a(n) = A000081(n) - (Sum_{1<=i<=j, i+j=n} A000081(i)*A000081(j)) + (1-(-1)^(n-1)) * binomial(A000081(n/2)+1,2) / 2 [Li, equation 4.2]. - Walt Rorie-Baety, Jul 05 2021

A000311 Schroeder's fourth problem; also series-reduced rooted trees with n labeled leaves; also number of total partitions of n.

Original entry on oeis.org

0, 1, 1, 4, 26, 236, 2752, 39208, 660032, 12818912, 282137824, 6939897856, 188666182784, 5617349020544, 181790703209728, 6353726042486272, 238513970965257728, 9571020586419012608, 408837905660444010496, 18522305410364986906624

Offset: 0

N. J. A. Sloane

a(n) is the number of labeled series-reduced rooted trees with n leaves (root has degree 0 or >= 2); a(n-1) = number of labeled series-reduced trees with n leaves. Also number of series-parallel networks with n labeled edges, divided by 2.
A total partition of n is essentially what is meant by the first part of the previous line: take the numbers 12...n, and partition them into at least two blocks. Partition each block with at least 2 elements into at least two blocks. Repeat until only blocks of size 1 remain. (See the reference to Stanley, Vol. 2.) - N. J. A. Sloane, Aug 03 2016
Polynomials with coefficients in triangle A008517, evaluated at 2. - Ralf Stephan, Dec 13 2004
Row sums of unsigned A134685. - Tom Copeland, Oct 11 2008
Row sums of A134991, which contains an e.g.f. for this sequence and its compositional inverse. - Tom Copeland, Jan 24 2018
From Gus Wiseman, Dec 28 2019: (Start)
Also the number of singleton-reduced phylogenetic trees with n labels. A phylogenetic tree is a series-reduced rooted tree whose leaves are (usually disjoint) nonempty sets. It is singleton-reduced if no non-leaf node covers only singleton branches. For example, the a(4) = 26 trees are:
  {1,2,3,4}  {{1},{2},{3,4}}  {{1},{2,3,4}}
             {{1},{2,3},{4}}  {{1,2},{3,4}}
             {{1,2},{3},{4}}  {{1,2,3},{4}}
             {{1},{2,4},{3}}  {{1,2,4},{3}}
             {{1,3},{2},{4}}  {{1,3},{2,4}}
             {{1,4},{2},{3}}  {{1,3,4},{2}}
                              {{1,4},{2,3}}
                              {{{1},{2,3}},{4}}
                              {{{1,2},{3}},{4}}
                              {{1},{{2},{3,4}}}
                              {{1},{{2,3},{4}}}
                              {{{1},{2,4}},{3}}
                              {{{1,2},{4}},{3}}
                              {{1},{{2,4},{3}}}
                              {{{1,3},{2}},{4}}
                              {{{1},{3,4}},{2}}
                              {{{1,3},{4}},{2}}
                              {{{1,4},{2}},{3}}
                              {{{1,4},{3}},{2}}
(End)

			E.g.f.: A(x) = x + x^2/2! + 4*x^3/3! + 26*x^4/4! + 236*x^5/5! + 2752*x^6/6! + ...
where exp(A(x)) = 1 - x + 2*A(x), and thus
Series_Reversion(A(x)) = x - x^2/2! - x^3/3! - x^4/4! - x^5/5! - x^6/6! + ...
O.g.f.: G(x) = x + x^2 + 4*x^3 + 26*x^4 + 236*x^5 + 2752*x^6 + 39208*x^7 + ...
where
G(x) = x/2 + x/(2*(2-x)) + x/(2*(2-x)*(2-2*x)) + x/(2*(2-x)*(2-2*x)*(2-3*x)) + x/(2*(2-x)*(2-2*x)*(2-3*x)*(2-4*x)) + x/(2*(2-x)*(2-2*x)*(2-3*x)*(2-4*x)*(2-5*x)) + ...
From _Gus Wiseman_, Dec 28 2019: (Start)
A rooted tree is series-reduced if it has no unary branchings, so every non-leaf node covers at least two other nodes. The a(4) = 26 series-reduced rooted trees with 4 labeled leaves are the following. Each bracket (...) corresponds to a non-leaf node.
  (1234)  ((12)34)  ((123)4)
          (1(23)4)  (1(234))
          (12(34))  ((124)3)
          (1(24)3)  ((134)2)
          ((13)24)  (((12)3)4)
          ((14)23)  ((1(23))4)
                    ((12)(34))
                    (1((23)4))
                    (1(2(34)))
                    (((12)4)3)
                    ((1(24))3)
                    (1((24)3))
                    (((13)2)4)
                    ((13)(24))
                    (((13)4)2)
                    ((1(34))2)
                    (((14)2)3)
                    ((14)(23))
                    (((14)3)2)
(End)

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 224.
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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).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see "total partitions", Example 5.2.5, Equation (5.27), and also Fig. 5-3 on page 14. See also the Notes on page 66.

N. J. A. Sloane, Table of n, a(n) for n = 0..375 [Shortened file because terms grow rapidly: see Sloane link below for additional terms]
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Tom Copeland, Comments on A000311
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Philippe Flajolet, A Problem in Statistical Classification Theory
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M. D. Hendy, C. H. C. Little, David Penny, Comparing trees with pendant vertices labelled, SIAM J. Appl. Math. 44 (5) (1984) Table 1
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 69
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T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176. [Annotated, scanned copy]
F. Murtagh, Counting dendrograms: a survey, Discrete Appl. Math., 7 (1984), 191-199.
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.
P. Regner, Phylogenetic Trees: Selected Combinatorial Problems, Master's Thesis, 2012, Institute of Discrete Mathematics and Geometry, TU Vienna, pp. 50-59.
J. Riordan, Letter to N. J. A. Sloane, Jul. 1970
J. Riordan, The blossoming of Schröder's fourth problem, Acta Math., 137 (1976), 1-16.
E. Schröder, Vier combinatorische Probleme, Z. f. Math. Phys., 15 (1870), 361-376. [Annotated scanned copy]
N. J. A. Sloane, Table of n, a(n) for n = 0..500
J. Taylor, Formal group laws and hypergraph colorings, doctoral thesis, Univ. of Wash., 2016, p. 95. [_Tom Copeland_, Dec 20 2018]
Elena L. Wang and Guoce Xin, On Ward Numbers and Increasing Schröder Trees, arXiv:2507.15654 [math.CO], 2025. See pp. 2, 12.
Index entries for sequences related to trees
Index entries for sequences related to rooted trees
Index entries for "core" sequences
Index entries for sequences mentioned in Moon (1987)
Index entries for sequences related to parenthesizing

Row sums of A064060 and A134991.
The unlabeled version is A000669.
Unlabeled phylogenetic trees are A141268.
The node-counting version is A060356, with unlabeled version A001678.
Phylogenetic trees with n labels are A005804.
Chains of set partitions are A005121, with maximal version A002846.
Inequivalent leaf-colorings of series-reduced rooted trees are A318231.
Cf. A001003, A007827, A005805, A006351, A000084.
For n >= 2, A000311(n) = A006351(n)/2 = A005640(n)/2^(n+1).
Cf. A000110, A000669 = unlabeled hierarchies, A119649.
Cf. A000670, A135494, A134685.
Cf. A048816, A196545, A213427, A316651, A316652, A330626, A330627.
Cf. A000169, A042977, A133314, A134685, A269939.

M:=499; a:=array(0..500); a[0]:=0; a[1]:=1; a[2]:=1; for n from 0 to 2 do lprint(n,a[n]); od: for n from 2 to M do a[n+1]:=(n+2)*a[n]+2*add(binomial(n,k)*a[k]*a[n-k+1],k=2..n-1); lprint(n+1,a[n+1]); od:
Order := 50; t1 := solve(series((exp(A)-2*A-1),A)=-x,A); A000311 := n-> n!*coeff(t1,x,n);
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(combinat[multinomial](n, n-i*j, i$j)/j!*
      a(i)^j*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> `if`(n<2, n, b(n, n-1)):
seq(a(n), n=0..40);  # Alois P. Heinz, Jan 28 2016
# faster program:
b:= proc(n, i) option remember;
    `if`(i=0 and n=0, 1, `if`(i<=0 or i>n, 0,
    i*b(n-1, i) + (n+i-1)*b(n-1, i-1))) end:
a:= n -> `if`(n<2, n, add(b(n-1, i), i=0..n-1)):
seq(a(n), n=0..40);  # Peter Luschny, Feb 15 2021

nn = 19; CoefficientList[ InverseSeries[ Series[1+2a-E^a, {a, 0, nn}], x], x]*Range[0, nn]! (* Jean-François Alcover, Jul 21 2011 *)
a[ n_] := If[ n < 1, 0, n! SeriesCoefficient[ InverseSeries[ Series[ 1 + 2 x - Exp[x], {x, 0, n}]], n]]; (* Michael Somos, Jun 04 2012 *)
a[n_] := (If[n < 2,n,(column = ConstantArray[0, n - 1]; column[[1]] = 1; For[j = 3, j <= n, j++, column = column * Flatten[{Range[j - 2], ConstantArray[0, (n - j) + 1]}] + Drop[Prepend[column, 0], -1] * Flatten[{Range[j - 1, 2*j - 3], ConstantArray[0, n - j]}];]; Sum[column[[i]], {i, n - 1}]  )]); Table[a[n], {n, 0, 20}] (* Peter Regner, Oct 05 2012, after a formula by Felsenstein (1978) *)
multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[multinomial[n, Join[{n-i*j}, Array[i&,j]]]/j!*a[i]^j *b[n-i*j, i-1], {j, 0, n/i}]]]; a[n_] := If[n<2, n, b[n, n-1]]; Table[ a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 07 2016, after Alois P. Heinz *)
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mtot[m_]:=Prepend[Join@@Table[Tuples[mtot/@p],{p,Select[sps[m],1Gus Wiseman, Dec 28 2019 *)
(* Lengthy but easy to follow *)
  lead[, n /; n < 3] := 0
  lead[h_, n_] := Module[{p, i},
        p = Position[h, {_}];
        Sum[MapAt[{#, n} &, h, p[[i]]], {i, Length[p]}]
        ]
  follow[h_, n_] := Module[{r, i},
        r = Replace[Position[h, {_}], {a__} -> {a, -1}, 1];
        Sum[Insert[h, n, r[[i]]], {i, Length[r]}]
        ]
  marry[, n /; n < 3] := 0
  marry[h_, n_] := Module[{p, i},
        p = Position[h, _Integer];
        Sum[MapAt[{#, n} &, h, p[[i]]], {i, Length[p]}]
        ]
  extend[a_ + b_, n_] := extend[a, n] + extend[b, n]
  extend[a_, n_] := lead[a, n] + follow[a, n] + marry[a, n]
  hierarchies[1] := hierarchies[1] = extend[hier[{}], 1]
  hierarchies[n_] := hierarchies[n] = extend[hierarchies[n - 1], n] (* Daniel Geisler, Aug 22 2022 *)

a(n):=if n=1 then 1 else sum((n+k-1)!*sum(1/(k-j)!*sum((2^i*(-1)^(i)*stirling2(n+j-i-1,j-i))/((n+j-i-1)!*i!),i,0,j),j,1,k),k,1,n-1); /* Vladimir Kruchinin, Jan 28 2012 */

{a(n) = local(A); if( n<0, 0, for( i=1, n, A = Pol(exp(A + x * O(x^i)) - A + x - 1)); n! * polcoeff(A, n))}; /* Michael Somos, Jan 15 2004 */

{a(n) = my(A); if( n<0, 0, A = O(x); for( i=1, n, A = intformal( 1 / (1 + x - 2*A))); n! * polcoeff(A, n))}; /* Michael Somos, Oct 25 2014 */

{a(n) = n! * polcoeff(serreverse(1+2*x - exp(x +x^2*O(x^n))), n)}
for(n=0, 30, print1(a(n), ", ")) \\ Paul D. Hanna, Oct 27 2014

\p100 \\ set precision
{A=Vec(sum(n=0, 600, 1.*x/prod(k=0, n, 2 - k*x + O(x^31))))}
for(n=0, 25, print1(if(n<1,0,round(A[n])),", ")) \\ Paul D. Hanna, Oct 27 2014

from functools import lru_cache
from math import comb
@lru_cache(maxsize=None)
def A000311(n): return n if n <= 1 else -(n-1)*A000311(n-1)+comb(n,m:=n+1>>1)*(0 if n&1 else A000311(m)**2) + (sum(comb(n,i)*A000311(i)*A000311(n-i) for i in range(1,m))<<1) # Chai Wah Wu, Nov 10 2022

E.g.f. A(x) satisfies exp A(x) = 2*A(x) - x + 1.
a(0)=0, a(1)=a(2)=1; for n >= 2, a(n+1) = (n+2)*a(n) + 2*Sum_{k=2..n-1} binomial(n, k)*a(k)*a(n-k+1).
a(1)=1; for n>1, a(n) = -(n-1) * a(n-1) + Sum_{k=1..n-1} binomial(n, k) * a(k) * a(n-k). - Michael Somos, Jun 04 2012
From the umbral operator L in A135494 acting on x^n comes, umbrally, (a(.) + x)^n = (n * x^(n-1) / 2) - (x^n / 2) + Sum_{j>=1} j^(j-1) * (2^(-j) / j!) * exp(-j/2) * (x + j/2)^n giving a(n) = 2^(-n) * Sum_{j>=1} j^(n-1) * ((j/2) * exp(-1/2))^j / j! for n > 1. - Tom Copeland, Feb 11 2008
Let h(x) = 1/(2-exp(x)), an e.g.f. for A000670, then the n-th term of A000311 is given by ((h(x)*d/dx)^n)x evaluated at x=0, i.e., A(x) = exp(x*a(.)) = exp(x*h(u)*d/du) u evaluated at u=0. Also, dA(x)/dx = h(A(x)). - Tom Copeland, Sep 05 2011 (The autonomous differential eqn. here is also on p. 59 of Jones. - Tom Copeland, Dec 16 2019)
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
a(n) = Sum_{k=1..n-1} (n+k-1)!*Sum_{j=1..k} (1/(k-j)!)*Sum_{i=0..j} 2^i*(-1)^i*Stirling2(n+j-i-1, j-i)/((n+j-i-1)!*i!), n>1, a(0)=0, a(1)=1. - Vladimir Kruchinin, Jan 28 2012
Using L. Comtet's identity and D. Wasserman's explicit formula for the associated Stirling numbers of second kind (A008299) one gets: a(n) = Sum_{m=1..n-1} Sum_{i=0..m} (-1)^i * binomial(n+m-1,i) * Sum_{j=0..m-i} (-1)^j * ((m-i-j)^(n+m-1-i))/(j! * (m-i-j)!). - Peter Regner, Oct 08 2012
G.f.: x/Q(0), where Q(k) = 1 - k*x - x*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 01 2013
G.f.: x*Q(0), where Q(k) = 1 - x*(k+1)/(x*(k+1) - (1-k*x)*(1-x-k*x)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 11 2013
a(n) ~ n^(n-1) / (sqrt(2) * exp(n) * (2*log(2)-1)^(n-1/2)). - Vaclav Kotesovec, Jan 05 2014
E.g.f. A(x) satisfies d/dx A(x) = 1 / (1 + x - 2 * A(x)). - Michael Somos, Oct 25 2014
O.g.f.: Sum_{n>=0} x / Product_{k=0..n} (2 - k*x). - Paul D. Hanna, Oct 27 2014
E.g.f.: (x - 1 - 2 LambertW(-exp((x-1)/2) / 2)) / 2. - Vladimir Reshetnikov, Oct 16 2015 (This e.g.f. is given in A135494, the entry alluded to in my 2008 formula, and in A134991 along with its compositional inverse. - Tom Copeland, Jan 24 2018)
a(0) = 0, a(1) = 1; a(n) = n! * [x^n] exp(Sum_{k=1..n-1} a(k)*x^k/k!). - Ilya Gutkovskiy, Oct 17 2017
a(n+1) = Sum_{k=0..n} A269939(n, k) for n >= 1. - Peter Luschny, Feb 15 2021

Name edited by Gus Wiseman, Dec 28 2019

A002064 Cullen numbers: a(n) = n*2^n + 1.

Original entry on oeis.org

1, 3, 9, 25, 65, 161, 385, 897, 2049, 4609, 10241, 22529, 49153, 106497, 229377, 491521, 1048577, 2228225, 4718593, 9961473, 20971521, 44040193, 92274689, 192937985, 402653185, 838860801, 1744830465, 3623878657, 7516192769, 15569256449, 32212254721, 66571993089

Offset: 0

N. J. A. Sloane

Binomial transform is A084859. Inverse binomial transform is A004277. - Paul Barry, Jun 12 2003
Let A be the Hessenberg matrix of order n defined by: A[1,j]=1, A[i,i]:=2,(i>1), A[i,i-1] =-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)= (-1)^(n-1)*coeff(charpoly(A,x),x). - Milan Janjic, Jan 26 2010
Indices of primes are listed in A005849. - M. F. Hasler, Jan 18 2015
Add the list of fractions beginning with 1/2 + 3/4 + 7/8 + ... + (2^n - 1)/2^n and take the sums pairwise from left to right. For 1/2 + 3/4 = 5/4, 5 + 4 = 9 = a(2); for 5/4 + 7/8 = 17/8, 17 + 8 = 25 = a(3); for 17/8 + 15/16 = 49/16, 49 + 16 = 65 = a(4); for 49/16 + 31/32 = 129/32, 129 + 32 = 161 = a(5). For each pairwise sum a/b, a + b = n*2^(n+1). - J. M. Bergot, May 06 2015
Number of divisors of (2^n)^(2^n). - Gus Wiseman, May 03 2021
Named after the Irish Jesuit priest James Cullen (1867-1933), who checked the primality of the terms up to n=100. - Amiram Eldar, Jun 05 2021

			G.f. = 1 + 3*x + 9*x^2 + 25*x^3 + 65*x^4 + 161*x^5 + 385*x^6 + 897*x^7 + ... - _Michael Somos_, Jul 18 2018

G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
R. K. Guy, Unsolved Problems in Number Theory, B20.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 240-242.
W. Sierpiński, Elementary Theory of Numbers. Państ. Wydaw. Nauk., Warsaw, 1964, p. 346.
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..300
Ray Ballinger, Cullen Primes: Definition and Status.
Attila Bérczes, István Pink, and Paul Thomas Young, Cullen numbers and Woodall numbers in generalized Fibonacci sequences, J. Num. Theor. (2024) Vol. 262, 86-102.
Yuri Bilu, Diego Marques, and Alain Togbé, Generalized Cullen numbers in linear recurrence sequences, Journal of Number Theory, Vol. 202 (2019), pp. 412-425; arXiv preprint, arXiv:1806.09441 [math.NT], 2018.
Daniel Birmajer, Juan B. Gil, David S. Kenepp, and Michael D. Weiner, Restricted generating trees for weak orderings, arXiv:2108.04302 [math.CO], 2021.
C. K. Caldwell, The Top Twenty: Cullen Primes.
James Cullen, Question 15897, Educational Times, Vol. 58 (December 1905), p. 534.
Orhan Eren and Yüksel Soykan, Gaussian Generalized Woodall Numbers, Arch. Current Res. Int'l (2023) Vol. 23, Iss. 8, Art. No. ACRI.108618, 48-68. See p. 50.
Orhan Eren and Yüksel Soykan, On Dual Hyperbolic Generalized Woodall Numbers, Archives Current Res. Int'l (2024) Vol. 24, Iss. 11, Art. No. ACRI.126420, 398-423. See p. 401.
Jon Grantham and Hester Graves, The abc Conjecture Implies That Only Finitely Many Cullen Numbers Are Repunits, arXiv:2009.04052 [math.NT], 2020.
José María Grau and Florian Luca, Cullen numbers with the Lehmer property, Proceedings of the American Mathematical Society, Vol. 140, No. 1 (2012), pp. 129-134; arXiv preprint, arXiv:1103.3578 [math.NT], Mar 18 2011.
Paul Leyland, Factors of Cullen and Woodall numbers.
Paul Leyland, Generalized Cullen and Woodall numbers.
Diego Marques, On Generalized Cullen and Woodall Numbers That are Also Fibonacci Numbers, Journal of Integer Sequences, Vol. 17 (2014), Article 14.9.4.
Hisanori Mishima, Factorizations of many number sequences, Cullen numbers (n = 1 to 100), (n = 101 to 200), (n = 201 to 300), (n = 301 to 323).
Simon Plouffe, Approximations de Séries Génératrices et Quelques Conjectures, Dissertation, Université du Québec à Montréal, 1992.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Wacław Sierpiński, Elementary Theory of Numbers, Warszawa 1964.
Amelia Carolina Sparavigna, On the generalized sums of Mersenne, Fermat, Cullen and Woodall Numbers, Politecnico di Torino (Italy, 2019).
Amelia Carolina Sparavigna, Composition Operations of Generalized Entropies Applied to the Study of Numbers, International Journal of Sciences, Vol. 8, No. 4 (2019), pp. 87-92.
Amelia Carolina Sparavigna, The groupoids of Mersenne, Fermat, Cullen, Woodall and other Numbers and their representations by means of integer sequences, Politecnico di Torino, Italy (2019), [math.NT].
Amelia Carolina Sparavigna, Some Groupoids and their Representations by Means of Integer Sequences, International Journal of Sciences, Vol. 8, No. 10 (2019).
Eric Weisstein's World of Mathematics, Cullen Number.
Wikipedia, Cullen number.
Index entries for linear recurrences with constant coefficients, signature (5,-8,4).

Cf. A005849, A003261, A050914, A130197, A134081, A001787, A143038, A156708, A181527.
Cf. A000325, A186947.
Diagonal k = n + 1 of A046688.
A000005 counts divisors of n.
A000312 = n^n.
A002109 gives hyperfactorials (sigma: A260146, omega: A303281).
A057156 = (2^n)^(2^n).
A062319 counts divisors of n^n.
A173339 lists positions of squares in A062319.
A188385 gives the highest prime exponent in n^n.
A249784 counts divisors of n^n^n.
Cf. A000169, A000272, A036289, A066959, A176029, A340841, A343656.

a002064 n = n * 2 ^ n + 1
a002064_list = 1 : 3 : (map (+ 1) $ zipWith (-) (tail xs) xs)
   where xs = map (* 4) a002064_list
-- Reinhard Zumkeller, Mar 16 2013

[n*2^n + 1: n in [0..40]]; // Vincenzo Librandi, May 07 2015

A002064:=-(1-2*z+2*z**2)/((z-1)*(-1+2*z)**2); # conjectured by Simon Plouffe in his 1992 dissertation

Table[n*2^n+1,{n,0,2*4!}] (* Vladimir Joseph Stephan Orlovsky, Apr 25 2010 *)
LinearRecurrence[{5,-8,4},{1,3,9},51] (* Harvey P. Dale, Oct 13 2011 *)
CoefficientList[Series[(1 - 2 x + 2 x^2)/((1 - x) (2 x - 1)^2), {x, 0, 50}], x] (* Vincenzo Librandi, May 07 2015 *)

A002064(n)=n*2^n+1  \\ M. F. Hasler, Oct 31 2012

a(n) = 4a(n-1) - 4a(n-2) + 1. - Paul Barry, Jun 12 2003
a(n) = sum of row (n+1) of triangle A130197. Example: a(3) = 25 = (12 + 8 + 4 + 1), row 4 of A130197. - Gary W. Adamson, May 16 2007
Row sums of triangle A134081. - Gary W. Adamson, Oct 07 2007
Equals row sums of triangle A143038. - Gary W. Adamson, Jul 18 2008
Equals row sums of triangle A156708. - Gary W. Adamson, Feb 13 2009
G.f.: -(1-2*x+2*x^2)/((-1+x)*(2*x-1)^2). a(n) = A001787(n+1)+1-A000079(n). - R. J. Mathar, Nov 16 2007
a(n) = 1 + 2^(n + log_2(n)) ~ 1 + A000079(n+A004257(n)). a(n) ~ A000051(n+A004257(n)). - Jonathan Vos Post, Jul 20 2008
a(0)=1, a(1)=3, a(2)=9, a(n) = 5*a(n-1)-8*a(n-2)+4*a(n-3). - Harvey P. Dale, Oct 13 2011
a(n) = A036289(n) + 1 = A003261(n) + 2. - Reinhard Zumkeller, Mar 16 2013
E.g.f.: 2*x*exp(2*x) + exp(x). - Robert Israel, Dec 12 2014
a(n) = 2^n * A000325(n) = 4^n * A186947(-n) for all n in Z. - Michael Somos, Jul 18 2018
a(n) = Sum_{i=0..n-1} a(i) + A000325(n+1). - Ivan N. Ianakiev, Aug 07 2019
a(n) = sigma((2^n)^(2^n)) = A000005(A057156(n)) = A062319(2^n). - Gus Wiseman, May 03 2021
Sum_{n>=0} 1/a(n) = A340841. - Amiram Eldar, Jun 05 2021

Edited by M. F. Hasler, Oct 31 2012

A007778 a(n) = n^(n+1).

Original entry on oeis.org

0, 1, 8, 81, 1024, 15625, 279936, 5764801, 134217728, 3486784401, 100000000000, 3138428376721, 106993205379072, 3937376385699289, 155568095557812224, 6568408355712890625, 295147905179352825856, 14063084452067724991009, 708235345355337676357632

Offset: 0

N. J. A. Sloane

Number of edges of the complete bipartite graph of order n+n^n, K_n,n^n. - Roberto E. Martinez II, Jan 07 2002
All rational solutions to the equation x^y = y^x, with x < y, are given by x = A000169(n+1)/A000312(n), y = A000312(n+1)/A007778(n), where n >= 1. - Nick Hobson, Nov 30 2006
a(n) is also the number of ways of writing an n-cycle as the product of n+1 transpositions. - Nikos Apostolakis, Nov 22 2008
a(n) is the total number of elements whose preimage is the empty set summed over all partial functions from [n] into [n]. - Geoffrey Critzer, Jan 12 2022

Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 67.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Nick Hobson, Exponential equation.
Yidong Sun and Jujuan Zhuang, lambda-factorials of n, arXiv:1007.1339 [math.CO], 2010. - _Peter Luschny_, Jul 09 2010

Cf. A000169, A000272, A000312, A007830, A008785, A008786, A008787, A008788, A008789, A008790, A008791, A135608.
Essentially the same as A065440.
Cf. A061250, A143857. [From Reinhard Zumkeller, Jul 23 2010]

[n^(n+1):n in [0..20]]; // Vincenzo Librandi, Jan 03 2012

seq( n^(n+1), n=0..20); # G. C. Greubel, Mar 05 2020

Table[n^(n+1), {n,0,20}] (* Vladimir Joseph Stephan Orlovsky, Oct 01 2008 *)

A007778[n]:=n^(n+1)$
makelist(A007778[n],n,0,30); /* Martin Ettl, Oct 29 2012 */

vector(21, n, my(m=n-1); m^(m+1)) \\ G. C. Greubel, Mar 05 2020

[n^(n+1) for n in (0..20)] # G. C. Greubel, Mar 05 2020

E.g.f.: -W(-x)/(1 + W(-x))^3, W(x) Lambert's function (principal branch).
a(n) = Sum_{k=0..n} binomial(n,k)*A000166(k+1)*(n+1)^(n-k). - Peter Luschny, Jul 09 2010
See A008517 and A134991 for similar e.g.f.s. and A048993. - Tom Copeland, Oct 03 2011
E.g.f.: d/dx {x/(T(x)*(1-T(x)))}, where T(x) = Sum_{n >= 1} n^(n-1)*x^n/n! is the tree function of A000169. - Peter Bala, Aug 05 2012
a(n) = n*A000312(n). - R. J. Mathar, Jan 12 2017
Sum_{n>=2} 1/a(n) = A135608. - Amiram Eldar, Nov 17 2020

A008517 Second-order Eulerian triangle T(n,k), 1 <= k <= n.

Original entry on oeis.org

1, 1, 2, 1, 8, 6, 1, 22, 58, 24, 1, 52, 328, 444, 120, 1, 114, 1452, 4400, 3708, 720, 1, 240, 5610, 32120, 58140, 33984, 5040, 1, 494, 19950, 195800, 644020, 785304, 341136, 40320, 1, 1004, 67260, 1062500, 5765500, 12440064, 11026296, 3733920, 362880

Offset: 1

N. J. A. Sloane

Second-order Eulerian numbers <> = T(n,k+1) count the permutations of the multiset {1,1,2,2,...,n,n} with k ascents with the restriction that for all m, all integers between the two copies of m are less than m. In particular, the two 1s are always next to each other.
When seen as coefficients of polynomials with descending exponents, evaluations are in A000311 (x=2) and A001662 (x=-1).
The row reversed triangle is A112007. There one can find comments on the o.g.f.s for the diagonals of the unsigned Stirling1 triangle |A008275|.
Stirling2(n,n-k) = Sum_{m=0..k-1} T(k,m+1)*binomial(n+k-1+m, 2*k), k>=1. See the Graham et al. reference p. 271 eq. (6.43).
This triangle is the coefficient triangle of the numerator polynomials appearing in the o.g.f. for the k-th diagonal (k >= 1) of the Stirling2 triangle A048993.
The o.g.f. for column k satisfies the recurrence G(k,x) = x*(2*x*(d/dx)G(k-1,x) + (2-k)*G(k-1,x))/(1-k*x), k >= 2, with G(1,x) = 1/(1-x). - Wolfdieter Lang, Oct 14 2005
This triangle is in some sense generated by the differential equation y' = 1 - 2/(1+x+y). (This is the differential equation satisfied by the function defined implicitly as x+y=exp(x-y).) If we take y = a(0) + a(1)x + a(2)x^2 + a(3)x^3 + ... and assume a(0)=c then all the a's may be calculated formally in terms of c and we have a(1) = (c-1)/(c+1) and, for n > 1, a(n) = 2^n/n! (1+c)^(1-2n)( T(n,1)c - T(n,2)c^2 + T(n,3)c^3 - ... + (-1)^(n-1) T(n,n)c^n ). - Moshe Shmuel Newman, Aug 08 2007
From the recurrence relation, the generating function F(x,y) := 1 + Sum_{n>=1, 1<=k<=n} [T(n,k)x^n/n!*y^k] satisfies the partial differential equation F = (1/y-2x)F_x + (y-1)F_y, with (non-elementary) solution F(x,y) = (1-y)/(1-Phi(w)) where w = y*exp(x(y-1)^2-y) and Phi(x) is defined by Phi(x) = x*exp(Phi(x)). By Lagrange inversion (see Wilf's book "generatingfunctionology", page 168, Example 1), Phi(x) = Sum_{n>=1} n^(n-1)*x^n/n!. Thus Phi(x) can alternatively be described as the e.g.f. for rooted labeled trees on n vertices A000169. - David Callan, Jul 25 2008
A method for solving PDEs such as the one above for F(x,y) is described in the Klazar reference (see pages 207-208). In his case, the auxiliary ODE dy/dx = b(x,y)/a(x,y) is exact; in this case it is not exact but has an integrating factor depending on y alone, namely y-1. The e.g.f. for the row sums A001147 is 1/sqrt(1-2*x) and the demonstration that F(x,1) = 1/sqrt(1-2*x) is interesting: two applications of l'Hopital's rule to lim_{y->1}F(x,y) yield F(x,1) = 1/(1-2x)*1/F(x,1). So l'Hopital's rule doesn't directly yield F(x,1) but rather an equation to be solved for F(x,1)!. - David Callan, Jul 25 2008
From Tom Copeland, Oct 12 2008; May 19 2010: (Start)
Let P(0,t)= 0, P(1,t)= 1, P(2,t)= t, P(3,t)= t + 2 t^2, P(4,t)= t + 8 t^2 + 6 t^3, ... be the row polynomials of the present array, then
exp(x*P(.,t)) = ( u + Tree(t*exp(u)) ) / (1-t) = WD(x*(1-t), t/(1-t)) / (1-t)
where u = x*(1-t)^2 - t, Tree(x) is the e.g.f. of A000169 and WD(x,t) is the e.g.f. for A134991, relating the Ward and 2-Eulerian polynomials by a simple transformation.
Note also apparently P(4,t) / (1-t)^3 = Ward Poly(4, t/(1-t)) = essentially an e.g.f. for A093500.
The compositional inverse of f(x,t) = exp(P(.,t)*x) about x=0 is
g(x,t) = ( x - (t/(1-t)^2)*(exp(x*(1-t))-x*(1-t)-1) )
= x - t*x^2/2! - t*(1-t)*x^3/3! - t*(1-t)^2*x^4/4! - t*(1-t)^3*x^5/5! - ... .
Can apply A134685 to these coefficients to generate f(x,t). (End)
Triangle A163936 is similar to the one given above except for an extra right hand column [1, 0, 0, 0, ... ] and that its row order is reversed. - Johannes W. Meijer, Oct 16 2009
From Tom Copeland, Sep 04 2011: (Start)
Let h(x,t) = 1/(1-(t/(1-t))*(exp(x*(1-t))-1)), an e.g.f. in x for row polynomials in t of A008292, then the n-th row polynomial in t of the table A008517 is given by ((h(x,t)*D_x)^(n+1))x with the derivative evaluated at x=0.
Also, df(x,t)/dx = h(f(x,t),t) where f(x,t) is an e.g.f. in x of the row polynomials in t of A008517, i.e., exp(x*P(.,t)) in Copeland's 2008 comment. (End)
The rows are the h-vectors of A134991. - Tom Copeland, Oct 03 2011
Hilbert series of the pre-WDVV ring, thus h-vectors of the Whitehouse simplicial complex (cf. Readdy, Table 1). - Tom Copeland, Sep 20 2014
Arises in Buckholtz's analysis of the error term in the series for exp(nz). - N. J. A. Sloane, Jul 05 2016

			Triangle begins:
  1;
  1,   2;
  1,   8,   6;
  1,  22,  58,  24;
  1,  52, 328, 444, 120; ...
Row 3: There are three plane increasing 0-1-2-3 trees on 3 vertices. The number of colors are shown to the right of a vertex.
.
    1o (2*t+1)         1o t*(t+2)      1o t*(t+2)
     |                 / \             / \
     |                /   \           /   \
    2o (2*t+1)      2o    3o        3o    2o
     |
     |
    3o
.
The total number of trees is (2*t+1)^2 + t*(t+2) + t*(t+2) = 1 + 8*t + 6*t^2.

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, 2nd ed. Addison-Wesley, Reading, MA, 1994, p. 270. [with offsets [0,0]: see A201637]

Vincenzo Librandi, Rows n = 1..50, flattened
Peter Bala, Diagonals of triangles with generating function exp(t*F(x)).
J. Fernando Barbero G., Jesús Salas, and Eduardo J. S. Villaseñor, Bivariate Generating Functions for a Class of Linear Recurrences. I. General Structure, arXiv:1307.2010 [math.CO], 2013.
J. F. Barbero G., J. Salas and E. J. S. Villaseñor, Bivariate Generating Functions for a Class of Linear Recurrences. II. Applications, arXiv preprint arXiv:1307.5624 [math.CO], 2013.
J. Fernando Barbero G., Jesús Salas, and Eduardo J. S. Villaseñor, Generalized Stirling permutations and forests: Higher-order Eulerian and Ward numbers, Electronic Journal of Combinatorics 22(3) (2015), #P3.37.
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.
J. D. Buckholtz, Concerning an approximation of Copson, Proc. Amer. Math. Soc., 14 (1963), 564-568.
Naiomi T. Cameron and Kendra Killpatrick, Statistics on Linear Chord Diagrams, arXiv:1902.09021 [math.CO], 2019.
L. Carlitz, The coefficients in an asymptotic expansion, Proc. Amer. Math. Soc. 16 (1965) 248-252.
L. Carlitz, Some numbers related to the Stirling numbers of the first and second kind, Univ. Beograd Publ. Elektrotehn. Fak. Ser. Mat. Fiz., Numbers 544-576 (1976): 49-55. [Annotated scanned copy. The triangle is A008517.]
T. Copeland, Generators, Inversion, and Matrix, Binomial, and Integral Transforms
R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey and D. E. Knuth, On the Lambert W Function, Advances in Computational Mathematics, Vol. 5 (1996), pp. 329-359, formula (3.6), alternative link.
Ming-Jian Ding and Jiang Zeng, Proof of an explicit formula for a series from Ramanujan's Notebooks via tree functions, arXiv:2307.00566 [math.CO], 2023.
D. Dominici, Nested derivatives: A simple method for computing series expansions of inverse functions. arXiv:math/0501052v2 [math.CA], 2005.
Sen-Peng Eu, Tung-Shan Fu, and Yeh-Jong Pan, A refined sign-balance of simsun permutations, Eur. J. Comb. 36 (2014) 97-109
W. Gautschi, Exponential integral ... for large values of n, Jrn. of Rsch. of the National Bureau of Standards, Vol. 62, No. 3, March 1959, Rsch. paper 2941. - _Tom Copeland_, Jan 02 2016
I. Gessel and R. P. Stanley, Stirling polynomials, J. Combin. Theory, A 24 (1978), 24-33.
J. Ginsburg, Note on Stirling's Numbers, Amer. Math. Monthly 35 (1928), no. 2, 77-80. - _David Callan_, Aug 27 2009
Jim Haglund and Mirko Visontai, Stable multivariate Eulerian polynomials and generalized Stirling permutations, European Journal of Combinatorics, Volume 33, Issue 4, May 2012, pp. 477-487.
M. Klazar, Twelve countings with rooted plane trees, European Journal of Combinatorics 18 (1997), 195-210; Addendum, 18 (1997), 739-740.
Dmitry V. Kruchinin and Vladimir V. Kruchinin, A generating function for the Euler numbers of the second kind and its application, arXiv:1802.09003 [math.CO], 2018.
Paul Levande, Two New Interpretations of the Fishburn Numbers and their Refined Generating Functions, arXiv:1006.3013 [math.CO], 2010.
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 29.
Peter Luschny, Second-order Eulerian numbers, A companion to A340556, Feb 2021.
Shi-Mei Ma, Some combinatorial sequences associated with context-free grammars, arXiv:1208.3104v2 [math.CO], 2012. - From _N. J. A. Sloane_, Aug 21 2012
S.-M. Ma, T. Mansour, and M. Schork. Normal ordering problem and the extensions of the Stirling grammar, arXiv preprint arXiv:1308.0169 [math.CO], 2013.
S.-M. Ma, T. Mansour, The 1/k-Eulerian polynomials and k-Stirling permutations, arXiv preprint arXiv:1409.6525 [math.CO], 2014.
O. J. Munch, Om potensproduktsummer [Norwegian, English summary], Nordisk Matematisk Tidskrift, 7 (1959), 5-19. [Annotated scanned copy]
O. J. Munch, Om potensproduktsummer [ Norwegian, English summary ], Nordisk Matematisk Tidskrift, 7 (1959), 5-19.
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.
M. A. Readdy, The pre-WDVV ring of physics and its topology, preprint 2002, The Ramanujan Journal, October 2005, Volume 10, Issue 2, pp 269-281.
Grzegorz Rzadkowski and M Urlinska, A Generalization of the Eulerian Numbers, arXiv preprint arXiv:1612.06635, 2016
C. D. Savage and G. Viswanathan, The 1/k-Eulerian polynomials, Elec. J. of Comb., Vol. 19, Issue 1, #P9 (2012). - From _N. J. A. Sloane_, Feb 06 2013
M. D. Schmidt, Generalized j-Factorial Functions, Polynomials, and Applications , J. Int. Seq. 13 (2010), 10.6.7, Section 5.1.
Umesh Shankar, Log-concavity of rows of triangular arrays satisfying a certain super-recurrence, arXiv:2508.12467 [math.CO], 2025. See p. 4.
L. M. Smiley, Completion of a rational function sequence of Carlitz, arXiv:0006106 [math.CO], 2000.
M. Z. Spivey, On Solutions to a General Combinatorial Recurrence, J. Int. Seq. 14 (2011) # 11.9.7.
Eric Weisstein's World of Mathematics, Second-Order Eulerian Triangle
Wikipedia, Eulerian numbers of the second kind

Columns include A005803, A004301, A006260.
Right-hand columns include A000142, A002538, A002539.
Row sums are A001147.
For a (0,0) based version as used in 'Concrete Mathematics' and by Maple see A201637. For a (0,0) based version which has this triangle as a subtriangle see A340556.

with(combinat): A008517 := proc(n, m) local k: add((-1)^(n+k)* binomial(2*n+1, k)* stirling1(2*n-m-k+1, n-m-k+1), k=0..n-m) end: seq(seq(A008517(n, m), m=1..n), n=1..8);
# Johannes W. Meijer, Oct 16 2009, revised Nov 22 2012
A008517 := proc(n,k) option remember; `if`(n=1,`if`(k=0,1,0), A008517(n-1,k)* (k+1) + A008517(n-1,k-1)*(2*n-k-1)) end: seq(print(seq(A008517(n,k), k=0..n-1)), n=1..9);
# Peter Luschny, Apr 20 2011

a[n_, m_] = Sum[(-1)^(n + k)*Binomial[2 n + 1, k]*StirlingS1[2n-m-k+1, n-m-k+1], {k, 0, n-m}]; Flatten[Table[a[n, m], {n, 1, 9}, {m, 1, n}]][[1 ;; 44]] (* Jean-François Alcover, May 18 2011, after Johannes W. Meijer *)

{T(n, k) = my(z); if( n<1, 0, z = 1 + O(x); for( k=1, n, z = 1 + intformal( z^2 * (z+y-1))); n! * polcoeff( polcoeff(z, n),k))}; /* Michael Somos, Oct 13 2002 */

{T(n,k)=polcoeff((1-x)^(2*n+1)*sum(j=0,2*n+1,j^(n+j)*x^j/j!*exp(-j*x +x*O(x^k))),k)} \\ Paul D. Hanna, Oct 31 2012
for(n=1,10,for(k=1,n,print1(T(n,k),", "));print(""))

T(n, m) = sum(k=0, n-m, (-1)^(n+k)*binomial(2*n+1, k)*stirling(2*n-m-k+1, n-m-k+1, 1)); \\ Michel Marcus, Dec 07 2021

@CachedFunction
def A008517(n, k):
    if n==1: return 1 if k==0 else 0
    return A008517(n-1,k)*(k+1)+A008517(n-1,k-1)*(2*n-k-1)
for n in (1..9): [A008517(n, k) for k in(0..n-1)] # Peter Luschny, Oct 31 2012

T(n,k) = 0 if n < k, T(1,1) = 1, T(n,-1) = 0, T(n,k) = k*T(n-1,k) + (2*n-k)*T(n-1,k-1).
a(n,m) = Sum_{k=0..n-m} (-1)^(n+k)*binomial(2*n+1, k)*Stirling1(2*n-m-k+1, n-m-k+1). - Johannes W. Meijer, Oct 16 2009
From Peter Bala, Sep 29 2011: (Start)
For k = 0,1,2,... put G(k,x,t) := x-(1+2^k*t)*x^2/2+(1+2^k*t+3^k*t^2)*x^3/3-(1+2^k*t+3^k*t^2+4^k*t^3)*x^4/4+.... Then the series reversion of G(k,x,t) with respect to x gives an e.g.f. for the present table when k = 1 and for the Eulerian numbers A008292 when k = 0.
Let v = -t*exp((1-t)^2*x-t) and let B(x,t) = -(1+1/t*LambertW(v))/(1+LambertW(v)). From the e.g.f. given by Copeland above we find B(x,t) = compositional inverse with respect to x of G(1,x,t) = Sum_{n>=1} R(n,t)*x^n/n! = x+(1+2*t)*x^2/2!+(1+8*t+6*t^2)*x^3/3!+.... The function B(x,t) satisfies the differential equation dB/dx = (1+B)*(1+t*B)^2 = 1 + (2*t+1)*B + t*(t+2)*B^2 + t^2*B^3.
Applying [Bergeron et al., Theorem 1] gives a combinatorial interpretation for the row generating polynomials R(n,t): R(n,t) counts plane increasing trees where each vertex has outdegree <= 3, the vertices of outdegree 1 come in 2*t+1 colors, the vertices of outdegree 2 come in t*(t+2) colors and the vertices of outdegree 3 come in t^2 colors. An example is given below. Cf. A008292. Applying [Dominici, Theorem 4.1] gives the following method for calculating the row polynomials R(n,t): Let f(x,t) = (1+x)*(1+t*x)^2 and let D be the operator f(x,t)*d/dx. Then R(n+1,t) = D^n(f(x,t)) evaluated at x = 0. (End)
From Tom Copeland, Oct 03 2011: (Start)
a(n,k) = Sum_{i=0..k} (-1)^(k-i) binomial(n-i,k-i) A134991(n,i), offsets 0.
  P(n+1,t) = (1-t)^(2n+1) Sum_{k>=1} k^(n+k) [t*exp(-t)]^k / k! for n>0; consequently, Sum_{k>=1} (-1)^k k^(n+k) x^k/k!= [1+LW(x)]^(-(2n+1))P[n+1,-LW(x)] where LW(x) is the Lambert W-Function and P(n,t), for n > 0, are the row polynomials as given in Copeland's 2008 comment. (End)
The e.g.f. A(x,t) = -v * { Sum_{j>=1} D(j-1,u) (-z)^j / j! } where u=x*(1-t)^2-t, v=(1+u)/(1-t), z=(t+u)/[(1+u)^2] and D(j-1,u) are the polynomials of A042977. dA(x,t)/dx=(1-t)/[1+u-(1-t)A(x,t)]=(1-t)/{1+LW[-t exp(u)]}, (Copeland's e.g.f. in 2008 comment). - Tom Copeland, Oct 06 2011
A133314 applied to the derivative of A(x,t) implies (a.+b.)^n = 0^n, for (b_n)=P(n+1,t) and (a_0)=1, (a_1)=-t, and (a_n)=-P(n,t) otherwise. E.g., umbrally, (a.+b.)^2 = a_2*b_0 + 2 a_1*b_1 + a_0*b_2 = 0. - Tom Copeland, Oct 08 2011
The compositional inverse (with respect to x) of y = y(t;x) = (x-t*(exp(x)-1)) is  1/(1-t)*y + t/(1-t)^3*y^2/2! + (t+2*t^2)/(1-t)^5*y^3/3! + (t+8*t^2+6*t^3)/(1-t)^7*y^4/4! + .... The numerator polynomials of the rational functions in t are the row polynomials of this triangle. As observed in the Comments section, the rational functions in t are the generating functions for the diagonals of the triangle of Stirling numbers of the second kind (A048993). See the Bala link for a proof. Cf. A112007 and A134991. - Peter Bala, Dec 04 2011
O.g.f. of row n: (1-x)^(2*n+1) * Sum_{k>=0} k^(n+k) * exp(-k*x) * x^k/k!. - Paul D. Hanna, Oct 31 2012
T(n, k) = n!*[x^n][t^k](egf) where egf = (1-t)/(1 + LambertW(-exp(t^2*x - 2*t*x - t + x)*t)) and after expansion W(-exp(-t)t) is substituted by (-t). - Shamil Shakirov, Feb 17 2025

A036740 a(n) = (n!)^n.

Original entry on oeis.org

1, 1, 4, 216, 331776, 24883200000, 139314069504000000, 82606411253903523840000000, 6984964247141514123629140377600000000, 109110688415571316480344899355894085582848000000000, 395940866122425193243875570782668457763038822400000000000000000000

Offset: 0

N. J. A. Sloane

(-1)^n*a(n) is the determinant of the n X n matrix m_{i,j} = T(n+i,j), 1 <= i,j <= n, where T(n,k) are the signed Stirling numbers of the first kind (A008275). Derived from methods given in Krattenthaler link. - Benoit Cloitre, Sep 17 2005
a(n) is also the number of binary operations on an n-element set which are right (or left) cancellative. These are also called right (left) cancellative magma or groupoids. The multiplication table of a right (left) cancellative magma is an n X n matrix with entries from an n element set such that the elements in each column (or row) are distinct. - W. Edwin Clark, Apr 09 2009
This sequence is mentioned in "Experimentation in Mathematics" as a sum-of-powers determinant. - John M. Campbell, May 07 2011
Determinant of the n X n matrix M_n = [m_n(i,j)] with m_n(i,j) = Stirling2(n+i,j) for 1<=i,j<=n. - Alois P. Heinz, Jul 26 2013

Jonathan Borwein, David Bailey and Roland Girgensohn, Experimentation in Mathematics: Computational Paths to Discovery, A K Peters, Ltd., 2004, p. 207.

Alois P. Heinz, Table of n, a(n) for n = 0..30
Christian Krattenthaler, Advanced determinant calculus, in: D. Foata and G. N. Han (eds.), The Andrews Festschrift, Springer, Berlin, Heidelberg, 2001, pp. 349-426; arXiv preprint, arXiv:math/9902004 [math.CO], 1999.

Cf. A000142, A000169, A000312.
Cf. A086687, A225764, A261114, A261490.
Main diagonal of A225816.

a:= n-> n!^n:
seq(a(n), n=0..12);  # Alois P. Heinz, Jul 25 2013

Table[(n!)^n,{n,0,10}] (* Harvey P. Dale, Sep 29 2013 *)

makelist(n!^n,n,0,10); /* Martin Ettl, Jan 13 2013 */

PARI
```
a(n)=n!^n;
```

a(n) = a(n-1)*n^n*(n-1)! = a(n-1)*A000169(n)*A000142(n) = A036740(n-1) * A000312(n)*A000142(n-1). - Henry Bottomley, Dec 06 2001
From Benoit Cloitre, Sep 17 2005: (Start)
a(n) = Product_{k=1..n} (k-1)!*k^k;
a(n) = A000178(n-1)*A002109(n) for n >= 1. (End)
a(n) ~ 2^(n/2) * Pi^(n/2) * n^(n*(2*n+1)/2) / exp(n^2-1/12). - Vaclav Kotesovec, Nov 14 2014
a(n) = Product_{k=1..n} k^n. - José de Jesús Camacho Medina, Jul 12 2016
Sum_{n>=0} 1/a(n) = A261114. - Amiram Eldar, Nov 16 2020

A075513 Triangle read by rows. T(n, m) are the coefficients of Sidi polynomials.

Original entry on oeis.org

1, -1, 2, 1, -8, 9, -1, 24, -81, 64, 1, -64, 486, -1024, 625, -1, 160, -2430, 10240, -15625, 7776, 1, -384, 10935, -81920, 234375, -279936, 117649, -1, 896, -45927, 573440, -2734375, 5878656, -5764801, 2097152, 1, -2048, 183708, -3670016, 27343750, -94058496, 161414428, -134217728, 43046721

Offset: 1

Wolfdieter Lang, Oct 02 2002

Coefficients of the Sidi polynomials (-1)^(n-1)*D_{n-1,1,n-1}(x), for n >=1, where D_{k,n,m}(z) is given in Theorem 4.2., p. 862, of Sidi [1980].
The row polynomials p(n, x) := Sum_{m=0..n-1} a(n, m)x^m, n >= 1, are obtained from ((Eu(x)^n)*(x-1)^n)/(n*x), where Eu(x) := xd/dx is the Euler-derivative with respect to x.
The row polynomials p(n, y) := Sum_{m=0..n-1} a(n, m)*y^m, n >= 1, are also obtained from ((d^m/dx^m)((exp(x)-1)^m)/m)/exp(x) after replacement of exp(x) by y. Here (d^m/dx^m)f(x), m >= 1, denotes m-fold differentiation of f(x) with respect to x.
b(k,m,n) := (Sum_{p=0..m-1} (a(m, p)*((p+1)*k)^n))/(m-1)!, n >= 0, has g.f. 1/Product_{p=1..m} (1 - k*p*x) for k = 1, 2,... and m = 1, 2,...
The (signed) row sums give A000142(n-1), n >= 1, (factorials) and (unsigned) A074932(n).
The (unsigned) columns give A000012 (powers of 1), 2*A001787(n+1), (3^2)*A027472(n), (4^3)*A038846(n-1), (5^4)*A036071(n-5), (6^5)*A036084(n-6), (7^6)*A036226(n-7), (8^7)*A053107(n-8) for m=0..7.
Right edge of triangle is A000169. - Michel Marcus, May 17 2013

			The triangle T(n, m)  begins:
  n\m 0     1      2        3        4         5         6          7       8
  1:  1
  2: -1     2
  3:  1    -8      9
  4: -1    24    -81       64
  5:  1   -64    486    -1024      625
  6: -1   160  -2430    10240   -15625      7776
  7:  1  -384  10935   -81920   234375   -279936    117649
  8: -1   896 -45927   573440 -2734375   5878656  -5764801    2097152
  9:  1 -2048 183708 -3670016 27343750 -94058496 161414428 -134217728 4304672
  [Reformatted by _Wolfdieter Lang_, Oct 12 2022]
-----------------------------------------------------------------------------
p(2,x) = -1+2*x = (1/(2*x))*x*(d/dx)*x*(d/dx)*(x-1)^2.

A. Sidi, Practical Extrapolation Methods: Theory and Applications, Cambridge University Press, Cambridge, 2003.

Wolfdieter Lang, On a Certain Family of Sidi Polynomials, May 2023.
Harlan J. Brothers, Pascal's triangle, Sidi polynomials, and powers of e, Missouri J. Math. Sci. (2025) Vol. 37, No. 1, 67-78.
Doron S. Lubinsky and Herbert Stahl, Some Explicit Biorthogonal Polynomials, (IN) Approximation Theory XI, (C.K. Chui, M. Neamtu, L. Schumaker, eds.), Nashboro Press, Nashville, 2005, pp. 279-285.
Avram Sidi, Numerical Quadrature and Nonlinear Sequence Transformations; Unified Rules for Efficient Computation of Integrals with Algebraic and Logarithmic Endpoint Singularities, Math. Comp., 35 (1980), 851-874.

Cf. A075510, A075511, A075512, A074932, A075515, A075516, A075906..A075925, A076002..A076013, A258773.

# Assuming offset 0.
seq(seq((-1)^(n-k)*binomial(n, k)*(k+1)^n, k=0..n), n=0..8);
# Alternative:
egf := x -> 1/(exp(LambertW(-exp(-x)*x*y) + x) - x*y):
ser := x -> series(egf(x), x, 12):
row := n -> seq(coeff(n!*coeff(ser(x), x, n), y, k), k=0..n):
seq(print(row(n)), n = 0..8); # Peter Luschny, Oct 21 2022

p[n_, x_] := p[n, x] = Nest[ x*D[#, x]& , (x-1)^n, n]/(n*x); a[n_, m_] := Coefficient[ p[n, x], x, m]; Table[a[n, m], {n, 1, 9}, {m, 0, n-1}] // Flatten (* Jean-François Alcover, Jul 03 2013 *)

tabl(nn) = {for (n=1, nn, for (m=0, n-1, print1((-1)^(n-m-1)*binomial(n-1, m)*(m+1)^(n-1), ", ");); print(););} \\ Michel Marcus, May 17 2013

T(n, m) = ((-1)^(n-m-1)) binomial(n-1, m)*(m+1)^(n-1), n >= m+1 >= 1, else 0.
G.f. for m-th column: ((m+1)^m)(x/(1+(m+1)*x))^(m+1), m >= 0.
E.g.f.: -LambertW(-x*y*exp(-x))/((1+LambertW(-x*y*exp(-x)))*x*y). - Vladeta Jovovic, Feb 13 2008 [corrected for offset 0 <= m <= n. For offset n >= 1 take the integral over x. - Wolfdieter Lang, Oct 12 2022]
T(n, k) = S(n, k+1) / n where S(, ) is triangle in A258773. - Michael Somos, May 13 2018
E.g.f. of column k, with offset n >= 0: exp(-(k + 1)*x)*((k + 1)*x)^k/k!. - Wolfdieter Lang, Oct 20 2022
E.g.f: 1/(exp(LambertW(-exp(-x)*x*y) + x) - x*y) assuming offset = 0. - Peter Luschny, Oct 21 2022

A129271 Number of labeled n-node connected graphs with at most one cycle.

Original entry on oeis.org

1, 1, 1, 4, 31, 347, 4956, 85102, 1698712, 38562309, 980107840, 27559801736, 849285938304, 28459975589311, 1030366840792576, 40079074477640850, 1666985134587145216, 73827334760713500233, 3468746291121007607808, 172335499299097826575564, 9027150377126199463936000

Offset: 0

Washington Bomfim, May 10 2008

The majority of those graphs of order 4 are trees since we have 16 trees and only 9 unicycles. See example.

Also connected graphs covering n vertices with at most n edges. The unlabeled version is A005703. - Gus Wiseman, Feb 19 2024

			a(4) = 16 + 3*3 = 31.
From _Gus Wiseman_, Feb 19 2024: (Start)
The a(0) = 1 through a(3) = 4 graph edge sets:
  {}  .  {{1,2}}  {{1,2},{1,3}}
                  {{1,2},{2,3}}
                  {{1,3},{2,3}}
                  {{1,2},{1,3},{2,3}}
(End)

J. Riordan, An Introduction to Combinatorial Analysis, Dover, 2002, p. 2.

Andrew Howroyd, Table of n, a(n) for n = 0..100
Wikipedia, PseudoForest.

Cf. A129137, A000272.
For any number of edges we have A001187, unlabeled A001349.
The unlabeled version is A005703.
The case of equality is A057500, covering A370317, cf. A370318.
The non-connected non-covering version is A133686.
The connected complement is A140638, unlabeled A140636, covering A367868.
The non-connected covering version is A367869 or A369191.
The version with loops is A369197, non-connected A369194.
A006125 counts graphs, A000088 unlabeled.
A006129 counts covering graphs, A002494 unlabeled.
A062734 counts connected graphs by number of edges.
Cf. A006649, A116508, A134964, A143543, A323818, A367862, A367863, A367867, A367916, A367917, A368951.

a := n -> `if`(n=0,1,((n-1)*exp(n)*GAMMA(n-1,n)+n^(n-2)*(3-n))/2):
seq(simplify(a(n)),n=0..16); # Peter Luschny, Jan 18 2016

nn=20;t=Sum[n^(n-1)x^n/n!,{n,1,nn}];Range[0,nn]!CoefficientList[Series[ Log[1/(1-t)]/2+t/2-3t^2/4+1,{x,0,nn}],x]  (* Geoffrey Critzer, Mar 23 2013 *)

seq(n)={my(t=-lambertw(-x + O(x*x^n))); Vec(serlaplace(log(1/(1-t))/2 + t/2 - 3*t^2/4 + 1))} \\ Andrew Howroyd, Nov 07 2019

a(0) = 1, for n >=1, a(n) = A000272(n) + A057500(n) = n^{n-2} + (n-1)(n-2)/2Sum_{r=1..n-2}( (n-3)!/(n-2-r)! )n^(n-2-r)
E.g.f.: log(1/(1-T(x)))/2 + T(x)/2 - 3*T(x)^2/4 + 1, where T(x) is the e.g.f. for A000169. - Geoffrey Critzer, Mar 23 2013
a(n) = ((n-1)*e^n*GAMMA(n-1,n)+n^(n-2)*(3-n))/2 for n>=1. - Peter Luschny, Jan 18 2016

Terms a(17) and beyond from Andrew Howroyd, Nov 07 2019

A001858 Number of forests of trees on n labeled nodes.

Original entry on oeis.org

1, 1, 2, 7, 38, 291, 2932, 36961, 561948, 10026505, 205608536, 4767440679, 123373203208, 3525630110107, 110284283006640, 3748357699560961, 137557910094840848, 5421179050350334929, 228359487335194570528, 10239206473040881277575, 486909744862576654283616

Offset: 0

N. J. A. Sloane and Simon Plouffe

The number of integer lattice points in the permutation polytope of {1,2,...,n}. - Max Alekseyev, Jan 26 2010
Equals the number of score sequences for a tournament on n vertices. See Prop. 7 of the article by Bartels et al., or Example 3.1 in the article by Stanley. - David Radcliffe, Aug 02 2022
Number of labeled acyclic graphs on n vertices. The unlabeled version is A005195. The covering case is A105784, connected A000272. - Gus Wiseman, Apr 29 2024

			From _Gus Wiseman_, Apr 29 2024: (Start)
Edge-sets of the a(4) = 38 forests:
  {}  {12}  {12,13}  {12,13,14}
      {13}  {12,14}  {12,13,24}
      {14}  {12,23}  {12,13,34}
      {23}  {12,24}  {12,14,23}
      {24}  {12,34}  {12,14,34}
      {34}  {13,14}  {12,23,24}
            {13,23}  {12,23,34}
            {13,24}  {12,24,34}
            {13,34}  {13,14,23}
            {14,23}  {13,14,24}
            {14,24}  {13,23,24}
            {14,34}  {13,23,34}
            {23,24}  {13,24,34}
            {23,34}  {14,23,24}
            {24,34}  {14,23,34}
                     {14,24,34}
(End)

B. Bollobas, Modern Graph Theory, Springer, 1998, p. 290.
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).

Alois P. Heinz, Table of n, a(n) for n = 0..388 (first 101 terms from T. D. Noe)
J. E. Bartels, J. Mount, and D. J. A. Welsh, The polytope of win vectors. Annals of Combinatorics 1, 1-15 (1997). https://doi.org/10.1007/BF02558460
David Callan, A Combinatorial Derivation of the Number of Labeled Forests, J. Integer Seqs., Vol. 6, 2003.
Huantian Cao, AutoGF: An Automated System to Calculate Coefficients of Generating Functions, thesis, 2002.o
Huantian Cao, AutoGF: An Automated System to Calculate Coefficients of Generating Functions, thesis, 2002.
Huantian Cao, AutoGF: An Automated System to Calculate Coefficients of Generating Functions, thesis, 2002 [Local copy, with permission]
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 132.
Samuele Giraudo, Combalgebraic structures on decorated cliques, Formal Power Series and Algebraic Combinatorics, Séminaire Lotharingien de Combinatoire, 78B.15, 2017, p. 7, arXiv:1709.08416 [math.CO], 2017.
Anton Izosimov, Matrix polynomials, generalized Jacobians, and graphical zonotopes, arXiv:1506.05179 [math.AG], 2015.
Qipeng Kuang, Ondřej Kuželka, Yuanhong Wang, and Yuyi Wang, Bridging Weighted First Order Model Counting and Graph Polynomials, arXiv:2407.11877 [cs.LO], 2024. See pp. 32-33.
Arun P. Mani and Rebecca J. Stones, Congruences for weighted number of labeled forests, INTEGERS 16 (2016). #A17.
J. Pitman, Coalescent Random Forests, J. Combin. Theory, A85 (1999), 165-193.
J. Riordan, Forests of labeled trees, J. Combin. Theory, 5 (1968), 90-103.
J. Riordan, Letter to N. J. A. Sloane, Oct 19 1970
J. Riordan, Letter to N. J. A. Sloane, Nov 10 1975
John Riordan and N. J. A. Sloane, Correspondence, 1974
N. J. A. Sloane, Letter to J. Riordan, Nov. 1970
R. Stanley, Decompositions of rational convex polytopes, Ann. Discrete Math 6.6 (1980): 333-342.
L. Takacs, On the number of distinct forests, SIAM J. Discrete Math., 3 (1990), 574-581.
E. M. Wright, A relationship between two sequences, Proc. London Math. Soc. (3) 17 (1967) 296-304.
Index entries for sequences related to forests

The connected case is A000272, rooted A000169.
The unlabeled version is A005195, connected A000055.
The covering case is A105784, unlabeled A144958.
Row sums of A138464.
For triangles instead of cycles we have A213434, covering A372168.
For a unique cycle we have A372193, covering A372195.
A002807 counts cycles in a complete graph.
A006125 counts simple graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
Cf. A006572, A006573, A088956.
Cf. A053530, A054548, A137916, A263340, A322661, A367863, A372176.

exp(x+x^2+add(n^(n-2)*x^n/n!, n=3..50));
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(
      binomial(n-1, j-1)*j^(j-2)*a(n-j), j=1..n))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Sep 15 2008
# third Maple program:
F:= exp(-LambertW(-x)*(1+LambertW(-x)/2)):
S:= series(F,x,51):
seq(coeff(S,x,j)*j!, j=0..50); # Robert Israel, May 21 2015

nn=20;t=Sum[n^(n-1)x^n/n!,{n,1,nn}];Range[0,nn]!CoefficientList[ Series[Exp[t-t^2/2],{x,0,nn}],x] (* Geoffrey Critzer, Sep 05 2012 *)
nmax = 20; CoefficientList[Series[-LambertW[-x]/(x*E^(LambertW[-x]^2/2)), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jul 19 2019 *)

a(n)=if(n<0,0,sum(m=0,n,sum(j=0,m,binomial(m,j)*binomial(n-1,n-m-j)*n^(n-m-j)*(m+j)!/(-2)^j)/m!)) /* Michael Somos, Aug 22 2002 */

E.g.f.: exp( Sum_{n>=1} n^(n-2)*x^n/n! ). This implies (by a theorem of Wright) that a(n) ~ exp(1/2)*n^(n-2). - N. J. A. Sloane, May 12 2008 [Corrected by Philippe Flajolet, Aug 17 2008]
E.g.f.: exp(T - T^2/2), where T = T(x) = Sum_{n>=1} n^(n-1)*x^n/n! is Euler's tree function (see A000169). - Len Smiley, Dec 12 2001
Shifts 1 place left under the hyperbinomial transform (cf. A088956). - Paul D. Hanna, Nov 03 2003
a(0) = 1, a(n) = Sum_{j=0..n-1} C(n-1,j) (j+1)^(j-1) a(n-1-j) if n>0. - Alois P. Heinz, Sep 15 2008

More terms from Michael Somos, Aug 22 2002

A052852 Expansion of e.g.f.: (x/(1-x))*exp(x/(1-x)).

Original entry on oeis.org

0, 1, 4, 21, 136, 1045, 9276, 93289, 1047376, 12975561, 175721140, 2581284541, 40864292184, 693347907421, 12548540320876, 241253367679185, 4909234733857696, 105394372192969489, 2380337795595885156, 56410454014314490981, 1399496554158060983080

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

A simple grammar.
Number of {121,212}-avoiding n-ary words of length n. - Ralf Stephan, Apr 20 2004
The infinite continued fraction (1+n)/(1+(2+n)/(2+(3+n)/(3+...))) converges to the rational number A052852(n)/A000262(n) when n is a positive integer. - David Angell (angell(AT)maths.unsw.edu.au), Dec 18 2008
a(n) is the total number of components summed over all nilpotent partial permutations of [n]. - Geoffrey Critzer, Feb 19 2022

Vincenzo Librandi, Table of n, a(n) for n = 0..200
David Angell, A family of continued fractions, Journal of Number Theory, Volume 130, Issue 4, April 2010, Pages 904-911, Section 2.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 820
Florent Hivert, Jean-Christophe Novelli and Jean-Yves Thibon, Commutative combinatorial Hopf algebras, arXiv:math/0605262 [math.CO], 2006.
John Riordan, Letter to N. J. A. Sloane, Sep 26 1980 with notes on the 1973 Handbook of Integer Sequences. Note that the sequences are identified by their N-numbers, not their A-numbers.
Luis Verde-Star, A Matrix Approach to Generalized Delannoy and Schröder Arrays, J. Int. Seq., Vol. 24 (2021), Article 21.4.1.
Michael Wallner, A bijection of plane increasing trees with relaxed binary trees of right height at most one, arXiv:1706.07163 [math.CO], 2017, Table 2 on p. 13.
Index entries for sequences related to Laguerre polynomials

Row sums of unsigned triangle A062139 (generalized a=2 Laguerre).
Cf. A000262. A000169, A000312.
Cf. A059114, A288268.

[n eq 0 select 0 else Factorial(n)*Evaluate(LaguerrePolynomial(n-1, 0), -1): n in [0..30]]; // G. C. Greubel, Feb 23 2021

spec := [S,{B=Set(C),C=Sequence(Z,1 <= card),S=Prod(B,C)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
a := n -> ifelse(n = 0, 0, n!*hypergeom([-n+1], [1], -1)): seq(simplify(a(n)), n = 0..18);  # Peter Luschny, Dec 30 2024

Table[n!*SeriesCoefficient[(x/(1-x))*E^(x/(1-x)),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 09 2012 *)
Table[If[n==0, 0, n!*LaguerreL[n-1, 0, -1]], {n, 0, 30}] (* G. C. Greubel, Feb 23 2021 *)

my(x='x+O('x^30)); concat([0], Vec(serlaplace((x/(1-x))*exp(x/(1-x))))) \\ G. C. Greubel, May 15 2018

[0 if n==0 else factorial(n)*gen_laguerre(n-1, 0, -1) for n in (0..30)] # G. C. Greubel, Feb 23 2021

D-finite with recurrence: a(1)=1, a(0)=0, (n^2+2*n)*a(n)+(-4-2*n)*a(n+1)+ a(n+2)=0.
a(n) = Sum_{m=0..n} n!*binomial(n+2, n-m)/m!. - Wolfdieter Lang, Jun 19 2001
a(n) = n*A002720(n-1). [Riordan] - Vladeta Jovovic, Mar 18 2005
Related to an n-dimensional series: for n>=1, a(n) = (n!/e)*Sum_{k_n>=k_{n-1}>=...>=k_1>=0} 1/(k_n)!. - Benoit Cloitre, Sep 30 2006
E.g.f.: (x/(1-x))*exp((x/(1-x))) = (x/(1-x))*G(0); G(k)=1+x/((2*k+1)*(1-x)-x*(1-x)*(2*k+1)/(x+(1-x)*(2*k+2)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Nov 24 2011
a(n) = D^n(x*exp(x)) evaluated at x = 0, where D is the operator (1+x)^2*d/dx. Cf. A000262 and A005493. - Peter Bala, Nov 25 2011
a(n) ~ exp(2*sqrt(n)-n-1/2)*n^(n+1/4)/sqrt(2). - Vaclav Kotesovec, Oct 09 2012
a(n) = n!*hypergeom([-n+1], [1], -1) for n>=1. - Peter Luschny, Oct 18 2014 [Simplified by Natalia L. Skirrow, 30 December 2024]
a(n) = Sum_{k=0..n} L(n,k)*k; L(n,k) the unsigned Lah numbers. - Peter Luschny, Oct 18 2014
a(n) = n!*LaguerreL(n-1, 0, -1) for n>=1. - Peter Luschny, Apr 08 2015, simplified Dec 30 2024
The series reversion of the e.g.f. equals W(x)/(1 + W(x)) = x - 2^2*x^2/2! + 3^3*x^3/3! - 4^4*x^4/4! + ..., essentially the e.g.f. for a signed version of A000312, where W(x) is Lambert's W-function (see A000169). - Peter Bala, Jun 14 2016
Equals column A059114(n, 2) for n >= 1. - G. C. Greubel, Feb 23 2021
a(n) = Sum_{k=1..n} k * A271703(n,k). - Geoffrey Critzer, Feb 19 2022

cp's OEIS Frontend

A000055 Number of trees with n unlabeled nodes.

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A000311 Schroeder's fourth problem; also series-reduced rooted trees with n labeled leaves; also number of total partitions of n.

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A002064 Cullen numbers: a(n) = n*2^n + 1.

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A007778 a(n) = n^(n+1).

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A008517 Second-order Eulerian triangle T(n,k), 1 <= k <= n.

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