A000312 -id:A000312 - cp's OEIS frontend

A000169 Number of labeled rooted trees with n nodes: n^(n-1).

1, 2, 9, 64, 625, 7776, 117649, 2097152, 43046721, 1000000000, 25937424601, 743008370688, 23298085122481, 793714773254144, 29192926025390625, 1152921504606846976, 48661191875666868481, 2185911559738696531968, 104127350297911241532841, 5242880000000000000000000

Offset: 1

N. J. A. Sloane

Also the number of connected transitive subtree acyclic digraphs on n vertices. - Robert Castelo, Jan 06 2001
For any given integer k, a(n) is also the number of functions from {1,2,...,n} to {1,2,...,n} such that the sum of the function values is k mod n. - Sharon Sela (sharonsela(AT)hotmail.com), Feb 16 2002
The n-th term of a geometric progression with first term 1 and common ratio n: a(1) = 1 -> 1,1,1,1,... a(2) = 2 -> 1,2,... a(3) = 9 -> 1,3,9,... a(4) = 64 -> 1,4,16,64,... - Amarnath Murthy, Mar 25 2004
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, 2, 3, ... . - Nick Hobson, Nov 30 2006
a(n+1) is also the number of partial functions on n labeled objects. - Franklin T. Adams-Watters, Dec 25 2006
In other words, if A is a finite set of size n-1, then a(n) is the number of binary relations on A that are also functions. Note that a(n) = Sum_{k=0..n-1} binomial(n-1,k)*(n-1)^k = n^(n-1), where binomial(n-1,k) is the number of ways to select a domain D of size k from A and (n-1)^k is the number of functions from D to A. - Dennis P. Walsh, Apr 21 2011
This is the fourth member of a set of which the other members are the symmetric group, full transformation semigroup, and symmetric inverse semigroup. For the first three, see A000142, A000312, A002720. - Peter J. Cameron, Nov 03 2024.
More generally, consider the class of sequences of the form a(n) = (n*c(1)*...*c(i))^(n-1). This sequence has c(1)=1. A052746 has a(n) = (2*n)^(n-1), A052756 has a(n) = (3*n)^(n-1), A052764 has a(n) = (4*n)^(n-1), A052789 has a(n) = (5*n)^(n-1) for n>0. These sequences have a combinatorial structure like simple grammars. - Ctibor O. Zizka, Feb 23 2008
a(n) is equal to the logarithmic transform of the sequence b(n) = n^(n-2) starting at b(2). - Kevin Hu (10thsymphony(AT)gmail.com), Aug 23 2010
Also, number of labeled connected multigraphs of order n without cycles except one loop. See link below to have a picture showing the bijection between rooted trees and multigraphs of this kind. (Note that there are no labels in the picture, but the bijection remains true if we label the nodes.) - Washington Bomfim, Sep 04 2010
a(n) is also the number of functions f:{1,2,...,n} -> {1,2,...,n} such that f(1) = 1.
For a signed version of A000169 arising from the Vandermonde determinant of (1,1/2,...,1/n), see the Mathematica section. - Clark Kimberling, Jan 02 2012
Numerator of (1+1/(n-1))^(n-1) for n>1. - Jean-François Alcover, Jan 14 2013
Right edge of triangle A075513. - Michel Marcus, May 17 2013
a(n+1) is the number of n x n binary matrices with no more than a single one in each row. Partitioning the set of such matrices by the number k of rows with a one, we obtain a(n+1) = Sum_{k=0..n} binomial(n,k)*n^k = (n+1)^n. - Dennis P. Walsh, May 27 2014
Central terms of triangle A051129: a(n) = A051129(2*n-1,n). - Reinhard Zumkeller, Sep 14 2014
a(n) is the row sum of the n-th rows of A248120 and A055302, so it enumerates the monomials in the expansion of [x(1) + x(2) + ... + x(n)]^(n-1). - Tom Copeland, Jul 17 2015
For any given integer k, a(n) is the number of sums x_1 + ... + x_m = k (mod n) such that: x_1, ..., x_m are nonnegative integers less than n, the order of the summands does not matter, and each integer appears fewer than n times as a summand. - Carlo Sanna, Oct 04 2015
a(n) is the number of words of length n-1 over an alphabet of n letters. - Joerg Arndt, Oct 07 2015
a(n) is the number of parking functions whose largest element is n and length is n. For example, a(3) = 9 because there are nine such parking functions, namely (1,2,3), (1,3,2), (2,3,1), (2,1,3), (3,1,2), (3,2,1), (1,1,3), (1,3,1), (3,1,1). - Ran Pan, Nov 15 2015
Consider the following problem: n^2 cells are arranged in a square array. A step can be defined as going from one cell to the one directly above it, to the right of it or under it. A step above cannot be followed by a step below and vice versa. Once the last column of the square array is reached, you can only take steps down. a(n) is the number of possible paths (i.e., sequences of steps) from the cell on the bottom left to the cell on the bottom right. - Nicolas Nagel, Oct 13 2016
The rationals c(n) = a(n+1)/a(n), n >= 1, appear in the proof of G. Pólya's "elementary, but not too elementary, theorem": Sum_{n>=1} (Product_{k=1..n} a_k)^(1/n) < exp(1)*Sum_{n>=1} a_n, for n >= 1, with the sequence {a_k}{k>=1} of nonnegative terms, not all equal to 0. - _Wolfdieter Lang, Mar 16 2018
Coefficients of the generating series for the preLie operadic algebra. Cf. p. 417 of the Loday et al. paper. - Tom Copeland, Jul 08 2018
a(n)/2^(n-1) is the square of the determinant of the n X n matrix M_n with elements m(j,k) = cos(Pi*j*k/n). See Zhi-Wei Sun, Petrov link. - Hugo Pfoertner, Sep 19 2021
a(n) is the determinant of the n X n matrix P_n such that, when indexed [0, n), P(0, j) = 1, P(i <= j) = i, and P(i > j) = i-n. - C.S. Elder, Mar 11 2024

			For n=3, a(3)=9 because there are exactly 9 binary relations on A={1, 2} that are functions, namely: {}, {(1,1)}, {(1,2)}, {(2,1)}, {(2,2)}, {(1,1),(2,1)}, {(1,1),(2,2)}, {(1,2),(2,1)} and {(1,2),(2,2)}. - _Dennis P. Walsh_, Apr 21 2011
G.f. = x + 2*x^2 + 9*x^3 + 64*x^4 + 625*x^5 + 7776*x^6 + 117649*x^7 + ...

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 169.
Jonathan L. Gross and Jay Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 524.
Hannes Heikinheimo, Heikki Mannila and Jouni K. Seppnen, Finding Trees from Unordered 01 Data, in Knowledge Discovery in Databases: PKDD 2006, Lecture Notes in Computer Science, Volume 4213/2006, Springer-Verlag. - N. J. A. Sloane, Jul 09 2009
Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 63.
John Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 128.
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).
Richard P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see page 25, Prop. 5.3.2, and p. 37, (5.52).

N. J. A. Sloane, Table of n, a(n) for n = 1..100
Paul Barry and Aoife Hennessy, Generalized Narayana Polynomials, Riordan Arrays, and Lattice Paths, Journal of Integer Sequences, Vol. 15, 2012, #12.4.8. - _N. J. A. Sloane_, Oct 08 2012
Washington Bomfim, Bijection between rooted forests and multigraphs without cycles except one loop in each connected component. [From _Washington Bomfim_, Sep 04 2010]
David Callan, A Combinatorial Derivation of the Number of Labeled Forests, J. Integer Seqs., Vol. 6, 2003.
Peter J. Cameron and Philippe Cara, Independent generating sets and geometries for symmetric groups, J. Algebra, Vol. 258, no. 2 (2002), 641-650.
Robert Castelo and Arno Siebes, A characterization of moral transitive directed acyclic graph Markov models as trees, Technical Report CS-2000-44, Faculty of Computer Science, University of Utrecht.
Robert Castelo and Arno Siebes, A characterization of moral transitive acyclic directed graph Markov models as labeled trees, Journal of Statistical Planning and Inference, Vol. 115, No. 1 (2003), pp. 235-259; alternative link.
Frédéric Chapoton, Florent Hivert and Jean-Christophe Novelli, A set-operad of formal fractions and dendriform-like sub-operads, Journal of Algebra, Vol. 465 (2016), pp. 322-355; arXiv preprint, arXiv:1307.0092 [math.CO], 2013.
Ali Chouria, Vlad-Florin Drǎgoi, Jean-Gabriel Luque, On recursively defined combinatorial classes and labelled trees, arXiv:2004.04203 [math.CO], 2020.
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; alternative link.
Nick Hobson, Solution to puzzle 48: Exponential equation.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 67.
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 p. 33.
Jean-Louis Loday and Bruno Vallette, Algebraic Operads, version 0.99, 2012.
G. Pólya, With, or Without, Motivation?, Amer. Math. Monthly, Vol. 56, No. 10 (1949), pp. 684-691. Reprinted in "A Century of Mathematics", John Ewing (ed.), Math. Assoc. of Amer., 1994, pp. 195-200 (the reference there is wrong).
Gwenaël Richomme, Characterization of infinite LSP words and endomorphisms preserving the LSP property, International Journal of Foundations of Computer Science, Vol. 30, No. 1 (2019), pp. 171-196; arXiv preprint, arXiv:1808.02680 [cs.DM], 2018.
Marko Riedel, math.stackexchange.com, Proof of an identity relating the tree function T(z) and the second order Eulerian numbers. Feb. 28, 2021.
Marko Riedel, math.stackexchange.com, Asymptotics of tree function statistics using Pusieux series
Frank Ruskey, Information on Rooted Trees.
N. J. A. Sloane, Illustration of initial terms
Zhi-Wei Sun, Fedor Petrov, A surprising identity, discussion in MathOverflow, Jan 17 2019.
Elena L. Wang and Guoce Xin, On Ward Numbers and Increasing Schröder Trees, arXiv:2507.15654 [math.CO], 2025. See p. 12.
Eric Weisstein's World of Mathematics, Graph Vertex.
Dimitri Zvonkine, An algebra of power series arising in the intersection theory of moduli spaces of curves and in the enumeration of ramified coverings of the sphere, arXiv:math/0403092 [math.AG], 2004.
Index entries for sequences related to rooted trees
Index entries for sequences related to trees
Index entries for "core" sequences

Cf. A000055, A000081, A000142, A000272, A000312, A002720, A007778, A007830, A008785-A008791, A055860, A002061, A052746, A052756, A052764, A052789, A051129, A098686, A247363, A055302, A248120, A130293, A053506-A053509, A262974.

Other classes of endofunctions: A275549-A275558.

a000169 n = n ^ (n - 1)  -- Reinhard Zumkeller, Sep 14 2014

[n^(n-1): n in [1..20]]; // Vincenzo Librandi, Jul 17 2015

A000169 := n -> n^(n-1);
# second program:
spec := [A, {A=Prod(Z, Set(A))}, labeled]; [seq(combstruct[count](spec, size=n), n=1..20)];
# third program:
A000169 := n -> add((-1)^(n+k-1)*pochhammer(n, k)*Stirling2(n-1, k), k = 0..n-1):
seq(A000169(n), n = 1 .. 23);  # Mélika Tebni, May 07 2023

Table[n^(n - 1), {n, 1, 20}] (* Stefan Steinerberger, Apr 01 2006 *)
Range[0, 18]! CoefficientList[ Series[ -LambertW[-x], {x, 0, 18}], x] // Rest (* Robert G. Wilson v, updated by Jean-François Alcover, Oct 14 2019 *)
(* Next, a signed version A000169 from the Vandermonde determinant of (1,1/2,...,1/n) *)
f[j_] := 1/j; z = 12;
v[n_] := Product[Product[f[k] - f[j], {j, 1, k - 1}], {k, 2, n}]
Table[v[n], {n, 1, z}]
1/%  (* A203421 *)
Table[v[n]/v[n + 1], {n, 1, z - 1}]  (* A000169 signed *)
(* Clark Kimberling, Jan 02 2012 *)
a[n_]:=Det[Table[If[i==0,1,If[i<=j,i,i-n]],{i,0,n-1},{j,0,n-1}]]; Array[a,20] (* Stefano Spezia, Mar 12 2024 *)

n^(n-1) $ n=1..20 /* Zerinvary Lajos, Apr 01 2007 */

PARI
```
a(n) = n^(n-1)
```

def a(n): return n**(n-1)
print([a(n) for n in range(1, 21)]) # Michael S. Branicky, Sep 19 2021

from sympy import Matrix
def P(n): return [[ (i-n if i > j else i) + (i == 0) for j in range(n) ] for i in range(n)]
print(*(Matrix(P(n)).det() for n in range(1, 21)), sep=', ') # C.S. Elder, Mar 12 2024

The e.g.f. T(x) = Sum_{n>=1} n^(n-1)*x^n/n! satisfies T(x) = x*exp(T(x)), so T(x) is the functional inverse (series reversion) of x*exp(-x).
Also T(x) = -LambertW(-x) where W(x) is the principal branch of Lambert's function.
T(x) is sometimes called Euler's tree function.
a(n) = A000312(n-1)*A128434(n,1)/A128433(n,1). - Reinhard Zumkeller, Mar 03 2007
E.g.f.: LambertW(x)=x*G(0); G(k) = 1 - x*((2*k+2)^(2*k))/(((2*k+1)^(2*k)) - x*((2*k+1)^(2*k))*((2*k+3)^(2*k+1))/(x*((2*k+3)^(2*k+1)) - ((2*k+2)^(2*k+1))/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Dec 30 2011
a(n) = Sum_{i=1..n} binomial(n-1,i-1)*i^(i-2)*(n-i)^(n-i). - Dmitry Kruchinin, Oct 28 2013
Limit_{n->oo} a(n)/A000312(n-1) = e. - Daniel Suteu, Jul 23 2016
From Amiram Eldar, Nov 20 2020: (Start)
Sum_{n>=1} 1/a(n) = A098686.
Sum_{n>=1} (-1)^(n+1)/a(n) = A262974. (End)
a(n) = Sum_{k=0..n-1} (-1)^(n+k-1)*Pochhammer(n, k)*Stirling2(n-1, k). - Mélika Tebni, May 07 2023
In terms of Eulerian numbers A340556(n,k) of the second order Sum_{m>=1} m^(m+n) z^m/m! = 1/(1-T(z))^(2n+1) * Sum_{k=0..n} A2(n,k) T(z)^k. - Marko Riedel, Jan 10 2024

A000272 Number of trees on n labeled nodes: n^(n-2) with a(0)=1.

Original entry on oeis.org

1, 1, 1, 3, 16, 125, 1296, 16807, 262144, 4782969, 100000000, 2357947691, 61917364224, 1792160394037, 56693912375296, 1946195068359375, 72057594037927936, 2862423051509815793, 121439531096594251776, 5480386857784802185939, 262144000000000000000000, 13248496640331026125580781

Offset: 0

N. J. A. Sloane

Number of spanning trees in complete graph K_n on n labeled nodes.
Robert Castelo, Jan 06 2001, observes that n^(n-2) is also the number of transitive subtree acyclic digraphs on n-1 vertices.
a(n) is also the number of ways of expressing an n-cycle in the symmetric group S_n as a product of n-1 transpositions, see example. - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 12 2001
Also counts parking functions, critical configurations of the chip firing game, allowable pairs sorted by a priority queue [Hamel].
The parking functions of length n can be described as all permutations of all words [d(1),d(2), ..., d(n)] where 1 <= d(k) <= k; see example. There are (n+1)^(n-1) = a(n+1) parking functions of length n. - Joerg Arndt, Jul 15 2014
a(n+1) is the number of endofunctions with no cycles of length > 1; number of forests of rooted labeled trees on n vertices. - Mitch Harris, Jul 06 2006
a(n) is also the number of nilpotent partial bijections (of an n-element set). Equivalently, the number of nilpotents in the partial symmetric semigroup, P sub n. - Abdullahi Umar, Aug 25 2008
a(n) is also the number of edge-labeled rooted trees on n nodes. - Nikos Apostolakis, Nov 30 2008
a(n+1) is the number of length n sequences on an alphabet of {1,2,...,n} that have a partial sum equal to n. For example a(4)=16 because there are 16 length 3 sequences on {1,2,3} in which the terms (beginning with the first term and proceeding sequentially) sum to 3 at some point in the sequence. {1, 1, 1}, {1, 2, 1}, {1, 2, 2}, {1, 2, 3}, {2, 1, 1}, {2, 1, 2}, {2, 1, 3}, {3, 1, 1}, {3, 1, 2}, {3, 1, 3}, {3, 2, 1}, {3, 2, 2}, {3, 2, 3}, {3, 3, 1}, {3, 3, 2}, {3, 3, 3}. - Geoffrey Critzer, Jul 20 2009
a(n) is the number of acyclic functions from {1,2,...,n-1} to {1,2,...,n}. An acyclic function f satisfies the following property: for any x in the domain, there exists a positive integer k such that (f^k)(x) is not in the domain. Note that f^k denotes the k-fold composition of f with itself, e.g., (f^2)(x)=f(f(x)). - Dennis P. Walsh, Mar 02 2011
a(n) is the absolute value of the discriminant of the polynomial x^{n-1}+...+x+1. More precisely, a(n) = (-1)^{(n-1)(n-2)/2} times the discriminant. - Zach Teitler, Jan 28 2014
For n > 2, a(n+2) is the number of nodes in the canonical automaton for the affine Weyl group of type A_n. - Tom Edgar, May 12 2016
The tree formula a(n) = n^(n-2) is due to Cayley (see the first comment). - Jonathan Sondow, Jan 11 2018
a(n) is the number of topologically distinct lines of play for the game Planted Brussels Sprouts on n vertices. See Ji and Propp link. - Caleb Ji, May 11 2018
a(n+1) is also the number of bases of R^n, that can be made from the n(n+1)/2 vectors of the form [0 ... 0 1 ... 1 0 ... 0]^T, where the initial or final zeros are optional, but at least one 1 has to be included. - Nicolas Nagel, Jul 31 2018
Cooper et al. show that every connected k-chromatic graph contains at least k^(k-2) spanning trees. - Michel Marcus, May 14 2020

			a(7)=matdet([196, 175, 140, 98, 56, 21; 175, 160, 130, 92, 53, 20; 140, 130, 110, 80, 47, 18; 98, 92, 80, 62, 38, 15; 56, 53, 47, 38, 26, 11; 21, 20, 18, 15, 11, 6])=16807
a(3)=3 since there are 3 acyclic functions f:[2]->[3], namely, {(1,2),(2,3)}, {(1,3),(2,1)}, and {(1,3),(2,3)}.
From _Joerg Arndt_ and Greg Stevenson, Jul 11 2011: (Start)
The following products of 3 transpositions lead to a 4-cycle in S_4:
  (1,2)*(1,3)*(1,4);
  (1,2)*(1,4)*(3,4);
  (1,2)*(3,4)*(1,3);
  (1,3)*(1,4)*(2,3);
  (1,3)*(2,3)*(1,4);
  (1,4)*(2,3)*(2,4);
  (1,4)*(2,4)*(3,4);
  (1,4)*(3,4)*(2,3);
  (2,3)*(1,2)*(1,4);
  (2,3)*(1,4)*(2,4);
  (2,3)*(2,4)*(1,2);
  (2,4)*(1,2)*(3,4);
  (2,4)*(3,4)*(1,2);
  (3,4)*(1,2)*(1,3);
  (3,4)*(1,3)*(2,3);
  (3,4)*(2,3)*(1,2).  (End)
The 16 parking functions of length 3 are 111, 112, 121, 211, 113, 131, 311, 221, 212, 122, 123, 132, 213, 231, 312, 321. - _Joerg Arndt_, Jul 15 2014
G.f. = 1 + x + x^2 + 3*x^3 + 16*x^4 + 125*x^5 + 1296*x^6 + 16807*x^7 + ...

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S. Janson, D. E. Knuth, T. Łuczak and B. Pittel, The Birth of the Giant Component, Random Structures and Algorithms Vol. 4 (1993), 233-358.
C. Ji and J. Propp, Brussels Sprouts, Noncrossing Trees, and Parking Functions, arXiv preprint arXiv:1805.03608 [math.CO], 2018.
L. Kalikow, Symmetries in trees and parking functions, Discrete Math., 256 (2002), 719-741.
C. Lamathe, The Number of Labelled k-Arch Graphs, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.1.
A. Laradji and A. Umar, On the number of nilpotents in the partial symmetric semigroup, Comm. Algebra 32 (2004), 3017-3023. From _Abdullahi Umar_, Aug 25 2008
C. J. Liu and Yutze Chow, On operator and formal sum methods for graph enumeration problems, SIAM J. Algebraic Discrete Methods, 5 (1984), no. 3, 384--406. MR0752043 (86d:05059). See Eq. (47). - From _N. J. A. Sloane_, Apr 09 2014
G. Martens, Polynomial Equations of Degree n, arXiv:math/0605183 [math.GM], 2006.
G. Martens, On Algebraic Solutions of Polynomial Equations of Degree n in one Variable, GH Consulting EPI-01-06 preprint, arXiv:math/0605183 [math.GM], 2006.
Mustafa A. A. Obaid, S. Khalid Nauman, Wafa S. Al Shammakh, Wafaa M. Fakieh, and Claus Michael Ringel, The number of complete exceptional sequences for a Dynkin algebra, arXiv preprint arXiv:1307.7573 [math.RT], 2013.
J. Pitman, Coalescent Random Forests, J. Combin. Theory, A85 (1999), 165-193.
J.-B. Priez and A. Virmaux, Non-commutative Frobenius characteristic of generalized parking functions: Application to enumeration, arXiv preprint arXiv:1411.4161 [math.CO], 2014.
S. Ramanujan, Question 738, J. Ind. Math. Soc.
J. Riordan, Forests of labeled trees, J. Combin. Theory, 5 (1968), 90-103. See Table 1.
M. P. Schützenberger, On an Enumeration Problem, Journal of Combinatorial Theory 4, 219-221 (1968). [A 1-1 correspondence between maps under permutations and acyclic maps.]
Alok Shukla, A short proof of Cayley's tree formula, Amer. Math. Monthly, 125 (2018), 65-68.
Stefano Spezia, A determinantal formula for the number of trees on n labeled nodes, 2023.
R. P. Stanley, Recent Progress in Algebraic Combinatorics, Bull. Amer. Math. Soc., 40 (2003), 55-68.
Dennis Walsh, Notes on acyclic functions and their directed graphs [Archived copy]
Eric Weisstein's World of Mathematics, Complete Graph, Labeled Tree, and Spanning Tree
D. Zeilberger, The n^(n-2)-th Proof Of The Formula For The Number Of Labeled Trees
D. Zeilberger, The n^(n-2)-th Proof Of The Formula For The Number Of Labeled Trees [Local copy]
D. Zeilberger, Yet Another Proof For The Enumeration Of Labeled Trees
D. Zeilberger, Yet Another Proof For The Enumeration Of Labeled Trees [Local copy]
D. Zvonkine, An algebra of power series arising in the intersection theory of moduli spaces of curves and in the enumeration of ramified coverings of the sphere, arXiv:math/0403092 [math.AG], 2004.
D. Zvonkine, Home Page [Dead link]
Index entries for sequences related to trees
Index entries for "core" sequences

Cf. A000169, A000254, A000312, A007778, A007830, A008785-A008791, A052750, A081048, A083483, A097170, A239910.
a(n) = A033842(n-1, 0) (first column of triangle).
a(n) = A058127(n-1, n) (right edge of triangle).
Cf. A000272 (labeled trees), A036361 (labeled 2-trees), A036362 (labeled 3-trees), A036506 (labeled 4-trees), A000055 (unlabeled trees), A054581 (unlabeled 2-trees).
Column m=1 of A105599. - Alois P. Heinz, Apr 10 2014

a000272 0 = 1; a000272 1 = 1
a000272 n = n ^ (n - 2)  -- Reinhard Zumkeller, Jul 07 2013

[ n^(n-2) : n in [1..10]]; // Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006

A000272 := n -> ifelse(n=0, 1, n^(n-2)): seq(A000272(n), n = 0..20); # Peter Luschny, Jun 12 2022

<< DiscreteMath`Combinatorica` Table[NumberOfSpanningTrees[CompleteGraph[n]], {n, 1, 20}] (* Artur Jasinski, Dec 06 2007 *)
Join[{1},Table[n^(n-2),{n,20}]] (* Harvey P. Dale, Nov 28 2012 *)
a[ n_] := If[ n < 1, Boole[n == 0], n^(n - 2)]; (* Michael Somos, May 25 2014 *)
a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ 1 - LambertW[-x] - LambertW[-x]^2 / 2, {x, 0, n}]]; (* Michael Somos, May 25 2014 *)
a[ n_] := If[ n < 1, Boole[n == 0], With[ {m = n - 1}, m! SeriesCoefficient[ Exp[ -LambertW[-x]], {x, 0, m}]]]; (* Michael Somos, May 25 2014 *)
a[ n_] := If[ n < 2, Boole[n >= 0], With[ {m = n - 1}, m! SeriesCoefficient[ InverseSeries[ Series[ Log[1 + x] / (1 + x), {x, 0, m}]], m]]]; (* Michael Somos, May 25 2014 *)
a[ n_] := If[ n < 1, Boole[n == 0], With[ {m = n - 1}, m! SeriesCoefficient[ Nest[ 1 + Integrate[ #^2 / (1 - x #), x] &, 1 + O[x], m], {x, 0, m}]]]; (* Michael Somos, May 25 2014 *)

A000272[n]:=if n=0 then 1 else n^(n-2)$
makelist(A000272[n],n,0,30); /* Martin Ettl, Oct 29 2012 */

{a(n) = if( n<1, n==0, n^(n-2))}; /* Michael Somos, Feb 16 2002 */

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

/* GP Function for Determinant of Hermitian (square symmetric) matrix for univariate polynomial of degree n by Gerry Martens: */
Hn(n=2)= {local(H=matrix(n-1,n-1),i,j); for(i=1,n-1, for(j=1,i, H[i,j]=(n*i^3-3*n*(n+1)*i^2/2+n*(3*n+1)*i/2+(n^4-n^2)/2)/6-(i^2-(2*n+1)*i+n*(n+1))*(j-1)*j/4; H[j,i]=H[i,j]; ); ); print("a(",n,")=matdet(",H,")"); print("Determinant H =",matdet(H)); return(matdet(H)); } { print(Hn(7)); } /* Gerry Martens, May 04 2007 */

def A000272(n): return 1 if n <= 1 else n**(n-2) # Chai Wah Wu, Feb 03 2022

E.g.f.: 1 + T - (1/2)*T^2; where T=T(x) is Euler's tree function (see A000169, also A001858). - Len Smiley, Nov 19 2001
Number of labeled k-trees on n nodes is binomial(n, k) * (k*(n-k)+1)^(n-k-2).
E.g.f. for b(n)=a(n+2): ((W(-x)/x)^2)/(1+W(-x)), where W is Lambert's function (principal branch). [Equals d/dx (W(-x)/(-x)). - Wolfdieter Lang, Oct 25 2022]
Determinant of the symmetric matrix H generated for a polynomial of degree n by: for(i=1,n-1, for(j=1,i, H[i,j]=(n*i^3-3*n*(n+1)*i^2/2+n*(3*n+1)*i/2+(n^4-n^2)/2)/6-(i^2-(2*n+1)*i+n*(n+1))*(j-1)*j/4; H[j,i]=H[i,j]; ); );. - Gerry Martens, May 04 2007
a(n+1) = Sum_{i=1..n} i * n^(n-1-i) * binomial(n, i). - Yong Kong (ykong(AT)curagen.com), Dec 28 2000
For n >= 1, a(n+1) = Sum_{i=1..n} n^(n-i)*binomial(n-1,i-1). - Geoffrey Critzer, Jul 20 2009
E.g.f. for b(n)=a(n+1): exp(-W(-x)), where W is Lambert's function satisfying W(x)*exp(W(x))=x. Proof is contained in link "Notes on acyclic functions..." - Dennis P. Walsh, Mar 02 2011
From Sergei N. Gladkovskii, Sep 18 2012: (Start)
E.g.f.: 1 + x + x^2/(U(0) - x) where U(k) = x*(k+1)*(k+2)^k + (k+1)^k*(k+2) - x*(k+2)^2*(k+3)*((k+1)*(k+3))^k/U(k+1); (continued fraction).
G.f.: 1 + x + x^2/(U(0)-x) where U(k) = x*(k+1)*(k+2)^k + (k+1)^k - x*(k+2)*(k+3)*((k+1)*(k+3))^k/E(k+1); (continued fraction). (End)
Related to A000254 by Sum_{n >= 1} a(n+1)*x^n/n! = series reversion( 1/(1 + x)*log(1 + x) ) = series reversion(x - 3*x^2/2! + 11*x^3/3! - 50*x^4/4! + ...). Cf. A052750. - Peter Bala, Jun 15 2016
For n >= 3 and 2 <= k <= n-1, the number of trees on n vertices with exactly k leaves is binomial(n,k)*S(n-2,n-k)(n-k)! where S(a,b) is the Stirling number of the second kind. Therefore a(n) = Sum_{k=2..n-1} binomial(n,k)*S(n-2,n-k)(n-k)! for n >= 3. - Jonathan Noel, May 05 2017

A051674 a(n) = prime(n)^prime(n).

Original entry on oeis.org

4, 27, 3125, 823543, 285311670611, 302875106592253, 827240261886336764177, 1978419655660313589123979, 20880467999847912034355032910567, 2567686153161211134561828214731016126483469, 17069174130723235958610643029059314756044734431

Offset: 1

Asher Auel

Numbers k such that bigomega(k)^(bigomega(k)) = k, where bigomega = A001222. - Lekraj Beedassy, Aug 21 2004
Positive k such that k' = k, where k' is the arithmetic derivative of k. - T. D. Noe, Oct 12 2004
David Beckwith proposes (in the AMM reference): "Let n be a positive integer and let p be a prime number. Prove that (p^p) | n! implies that (p^(p + 1)) | n!". - Jonathan Vos Post, Feb 20 2006
Subsequence of A100716; A003415(m*a(n)) = A129283(m)*a(n), especially A003415(a(n)) = a(n). - Reinhard Zumkeller, Apr 07 2007
A168036(a(n)) = 0. - Reinhard Zumkeller, May 22 2015

			a(1) = 2^2 = 4.
a(2) = 3^3 = 27.
a(3) = 5^5 = 3125.

J.-M. De Koninck & A. Mercier, 1001 Problemes en Theorie Classique Des Nombres, Problem 740 pp. 95; 312, Ellipses Paris 2004.

T. D. Noe, Table of n, a(n) for n = 1..40
David Beckwith, Problem 11158, American Mathematical Monthly, Vol. 112, No. 5 (May 2005), p. 468.
Jurij Kovic, The Arithmetic Derivative and Antiderivative, Journal of Integer Sequences, Vol. 15 (2012), #12.3.8.

Cf. A000040, A000312, A003415 (arithmetic derivative of n), A129150, A129151, A129152, A048102, A072873 (multiplicative closure), A104126.
Subsequence of A100717; A203908(a(n)) = 0.
Subsequence of A097764.
Cf. A168036, A094289 (decimal expansion of Sum(1/p^p)).

a051674_list = map (\p -> p ^ p) a000040_list
-- Reinhard Zumkeller, Jan 21 2012

[p^p: p in PrimesUpTo(30)]; // Vincenzo Librandi, Mar 27 2014

A051674:=n->ithprime(n)^ithprime(n): seq(A051674(n), n=1..10); # Wesley Ivan Hurt, Jun 25 2016

Array[Prime[ # ]^Prime[ # ] &, 12] (* Vladimir Joseph Stephan Orlovsky, May 01 2008 *)
#^#&/@Prime[Range[10]] (* Harvey P. Dale, May 17 2024 *)

a(n)=n=prime(n);n^n \\ Charles R Greathouse IV, Mar 20 2013

from gmpy2 import mpz
[mpz(prime(n))**mpz(prime(n)) for n in range(1,100)] # Chai Wah Wu, Jul 28 2014

a(n) = A000312(A000040(n)). - Altug Alkan, Sep 01 2016

Sum_{n>=1} 1/a(n) = A094289. - Amiram Eldar, Oct 13 2020

A002109 Hyperfactorials: Product_{k = 1..n} k^k.

Original entry on oeis.org

1, 1, 4, 108, 27648, 86400000, 4031078400000, 3319766398771200000, 55696437941726556979200000, 21577941222941856209168026828800000, 215779412229418562091680268288000000000000000, 61564384586635053951550731889313964883968000000000000000

Offset: 0

N. J. A. Sloane

A054374 gives the discriminants of the Hermite polynomials in the conventional (physicists') normalization, and A002109 (this sequence) gives the discriminants of the Hermite polynomials in the (in my opinion more natural) probabilists' normalization. See refs Wikipedia and Szego, eq. (6.71.7). - Alan Sokal, Mar 02 2012
a(n) = (-1)^n/det(M_n) where M_n is the n X n matrix m(i,j) = (-1)^i/i^j. - Benoit Cloitre, May 28 2002
a(n) = determinant of the n X n matrix M(n) where m(i,j) = B(n,i,j) and B(n,i,x) denote the Bernstein polynomial: B(n,i,x) = binomial(n,i)*(1-x)^(n-i)*x^i. - Benoit Cloitre, Feb 02 2003
Partial products of A000312. - Reinhard Zumkeller, Jul 07 2012
Number of trailing zeros (A246839) increases every 5 terms since the exponent of the factor 5 increases every 5 terms and the exponent of the factor 2 increases every 2 terms. - Chai Wah Wu, Sep 03 2014
Also the number of minimum distinguishing labelings in the n-triangular honeycomb rook graph. - Eric W. Weisstein, Jul 14 2017
Also shows up in a term in the solution to the generalized version of Raabe's integral. - Jibran Iqbal Shah, Apr 24 2021

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 135-145.
A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 50.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 477.
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).
G. Szego, Orthogonal Polynomials, American Mathematical Society, 1981 edition, 432 Pages.

N. J. A. Sloane, Table of n, a(n) for n = 0..36
Christian Aebi and Grant Cairns, Generalizations of Wilson's Theorem for Double-, Hyper-, Sub-and Superfactorials, The American Mathematical Monthly 122.5 (2015): 433-443.
Mohammad K. Azarian, On the Hyperfactorial Function, Hypertriangular Function, and the Discriminants of Certain Polynomials, International Journal of Pure and Applied Mathematics 36(2), 2007, pp. 251-257. MR2312537. Zbl 1133.11012.
blackpenredpen, What is a Hyperfactorial? Youtube video (2018).
CreativeMathProblems, A beautiful integral | Raabe's integral, Youtube Video (2021).
Steven R. Finch, Glaisher-Kinkelin Constant (gives asymptotic expressions for A002109, A000178) [Broken link]
Steven R. Finch, Glaisher-Kinkelin Constant (gives asymptotic expressions for A002109, A000178) [From the Wayback machine]
Shyam Sunder Gupta, Fascinating Factorials, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 16, 411-442.
A. M. Ibrahim, Extension of factorial concept to negative numbers, Notes on Number Theory and Discrete Mathematics, Vol. 19, 2013, 2, 30-42.
Jeffrey C. Lagarias and Harsh Mehta, Products of binomial coefficients and unreduced Farey fractions, arXiv:1409.4145 [math.NT], 2014.
Jean-Christophe Pain, Series representations for the logarithm of the Glaisher-Kinkelin constant, arXiv:2304.07629 [math.NT], 2023.
Jean-Christophe Pain, Bounds on the p-adic valuation of the factorial, hyperfactorial and superfactorial, arXiv:2408.00353 [math.NT], 2024. See p. 5.
Vignesh Raman, The Generalized Superfactorial, Hyperfactorial and Primorial functions, arXiv:2012.00882 [math.NT], 2020.
Jonathan Sondow and Petros Hadjicostas, The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant, J. Math. Anal. Appl., 332 (2007), 292-314; see Section 5.
László Tóth, Weighted gcd-sum functions, J. Integer Sequences, 14 (2011), Article 11.7.7.
Eric Weisstein's World of Mathematics, Hyperfactorial.
Eric Weisstein's World of Mathematics, K-Function.
Wikipedia, Hermite polynomials.
Index entries for sequences related to factorial numbers.
Index to divisibility sequences.

Cf. A000178, A000142, A000312, A001358, A002981, A002982, A100015, A005234, A014545, A018239, A006794, A057704, A057705, A054374.
Cf. A074962 [Glaisher-Kinkelin constant, also gives an asymptotic approximation for the hyperfactorials].
Cf. A001923, A051675, A240993, A255321, A255323, A255344.
Cf. A246839 (trailing 0's).
Cf. A347345, A347352.
Cf. A261175 (number of digits).

a002109 n = a002109_list !! n
a002109_list = scanl1 (*) a000312_list  -- Reinhard Zumkeller, Jul 07 2012

f := proc(n) local k; mul(k^k,k=1..n); end;
A002109 := n -> exp(Zeta(1,-1,n+1)-Zeta(1,-1));
seq(simplify(A002109(n)),n=0..11); # Peter Luschny, Jun 23 2012

Table[Hyperfactorial[n], {n, 0, 11}] (* Zerinvary Lajos, Jul 10 2009 *)
Hyperfactorial[Range[0, 11]] (* Eric W. Weisstein, Jul 14 2017 *)
Join[{1},FoldList[Times,#^#&/@Range[15]]] (* Harvey P. Dale, Nov 02 2023 *)

a(n)=prod(k=2,n,k^k) \\ Charles R Greathouse IV, Jan 12 2012

a(n)=polcoeff(1-sum(k=0, n-1, a(k)*x^k/prod(j=1,k+1,(1+j^j*x+x*O(x^n)) )), n) \\ Paul D. Hanna, Oct 02 2013

A002109 = [1]
for n in range(1, 10):
    A002109.append(A002109[-1]*n**n) # Chai Wah Wu, Sep 03 2014

a = lambda n: prod(falling_factorial(n,k) for k in (1..n))
[a(n) for n in (0..10)]  # Peter Luschny, Nov 29 2015

a(n)*A000178(n-1) = (n!)^n = A036740(n) for n >= 1.
Determinant of n X n matrix m(i, j) = binomial(i*j, i). - Benoit Cloitre, Aug 27 2003
a(n) = exp(zeta'(-1, n + 1) - zeta'(-1)) where zeta(s, z) is the Hurwitz zeta function. - Peter Luschny, Jun 23 2012
G.f.: 1 = Sum_{n>=0} a(n)*x^n / Product_{k=1..n+1} (1 + k^k*x). - Paul D. Hanna, Oct 02 2013
a(n) = A240993(n) / A000142(n+1). - Reinhard Zumkeller, Aug 31 2014
a(n) ~ A * n^(n*(n+1)/2 + 1/12) / exp(n^2/4), where A = 1.2824271291006226368753425... is the Glaisher-Kinkelin constant (see A074962). - Vaclav Kotesovec, Feb 20 2015
a(n) = Product_{k=1..n} ff(n,k) where ff denotes the falling factorial. - Peter Luschny, Nov 29 2015
log a(n) = (1/2) n^2 log n - (1/4) n^2 + (1/2) n log n + (1/12) log n + log(A) + o(1), where log(A) = A225746 is the logarithm of Glaisher's constant. - Charles R Greathouse IV, Mar 27 2020
From Amiram Eldar, Apr 30 2023: (Start)
Sum_{n>=1} 1/a(n) = A347345.
Sum_{n>=1} (-1)^(n+1)/a(n) = A347352. (End)
From Andrea Pinos, Apr 04 2024: (Start)
a(n) = e^(Integral_{x=1..n+1} (x - 1/2 - log(sqrt(2*Pi)) + (n+1-x)*Psi(x)) dx), where Psi(x) is the digamma function.
a(n) = e^(Integral_{x=1..n} (x + 1/2 - log(sqrt(2*Pi)) + log(Gamma(x+1))) dx). (End)

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

cp's OEIS Frontend

A109391 a(n) = (n^(n+1))*(n + 1)/2 = A000217(n)*A000312(n).

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A273068 Discriminator of the sequence n^n (A000312).

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A309377 a(n) is the product of the divisors of n^n (A000312).

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A349874 Expansion of A(x)^3, where A(x) is g.f. of n^n (A000312).

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A000169 Number of labeled rooted trees with n nodes: n^(n-1).

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A000272 Number of trees on n labeled nodes: n^(n-2) with a(0)=1.

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A051674 a(n) = prime(n)^prime(n).

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A109391 a(n) = (n^(n+1))(n + 1)/2 = A000217(n)A000312(n).