A128433 -id:A128433 - cp's OEIS frontend

A000312 a(n) = n^n; number of labeled mappings from n points to themselves (endofunctions).

1, 1, 4, 27, 256, 3125, 46656, 823543, 16777216, 387420489, 10000000000, 285311670611, 8916100448256, 302875106592253, 11112006825558016, 437893890380859375, 18446744073709551616, 827240261886336764177, 39346408075296537575424, 1978419655660313589123979

Offset: 0

N. J. A. Sloane

Also number of labeled pointed rooted trees (or vertebrates) on n nodes.
For n >= 1 a(n) is also the number of n X n (0,1) matrices in which each row contains exactly one entry equal to 1. - Avi Peretz (njk(AT)netvision.net.il), Apr 21 2001
Also the number of labeled rooted trees on (n+1) nodes such that the root is lower than its children. Also the number of alternating labeled rooted ordered trees on (n+1) nodes such that the root is lower than its children. - Cedric Chauve (chauve(AT)lacim.uqam.ca), Mar 27 2002
With p(n) = the number of integer partitions of n, p(i) = the number of parts of the i-th partition of n, d(i) = the number of different parts of the i-th partition of n, p(j, i) = the j-th part of the i-th partition of n, m(i, j) = multiplicity of the j-th part of the i-th partition of n, one has: a(n) = Sum_{i=1..p(n)} (n!/(Product_{j=1..p(i)} p(i, j)!)) * ((n!/(n - p(i)))!/(Product_{j=1..d(i)} m(i, j)!)). - Thomas Wieder, May 18 2005
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) is the total number of leaves in all (n+1)^(n-1) trees on {0,1,2,...,n} rooted at 0. For example, with edges directed away from the root, the trees on {0,1,2} are {0->1,0->2},{0->1->2},{0->2->1} and contain a total of a(2)=4 leaves. - David Callan, Feb 01 2007
Limit_{n->infinity} A000169(n+1)/a(n) = exp(1). Convergence is slow, e.g., it takes n > 74 to get one decimal place correct and n > 163 to get two of them. - Alonso del Arte, Jun 20 2011
Also smallest k such that binomial(k, n) is divisible by n^(n-1), n > 0. - Michel Lagneau, Jul 29 2013
For n >= 2 a(n) is represented in base n as "one followed by n zeros". - R. J. Cano, Aug 22 2014
Number of length-n words over the alphabet of n letters. - Joerg Arndt, May 15 2015
Number of prime parking functions of length n+1. - Rui Duarte, Jul 27 2015
The probability density functions p(x, m=q, n=q, mu=1) = A000312(q)*E(x, q, q) and p(x, m=q, n=1, mu=q) = (A000312(q)/A000142(q-1))*x^(q-1)*E(x, q, 1), with q >= 1, lead to this sequence, see A163931, A274181 and A008276. - Johannes W. Meijer, Jun 17 2016
Satisfies Benford's law [Miller, 2015]. - N. J. A. Sloane, Feb 12 2017
A signed version of this sequence apart from the first term (1, -4, -27, 256, 3125, -46656, ...), has the following property: for every prime p == 1 (mod 2n), (-1)^(n(n-1)/2)*n^n = A057077(n)*a(n) is always a 2n-th power residue modulo p. - Jianing Song, Sep 05 2018
From Juhani Heino, May 07 2019: (Start)
n^n is both Sum_{i=0..n} binomial(n,i)*(n-1)^(n-i)
  and Sum_{i=0..n} binomial(n,i)*(n-1)^(n-i)*i.
The former is the familiar binomial distribution of a throw of n n-sided dice, according to how many times a required side appears, 0 to n. The latter is the same but each term is multiplied by its amount. This means that if the bank pays the player 1 token for each die that has the chosen side, it is always a fair game if the player pays 1 token to enter - neither bank nor player wins on average.
Examples:
2-sided dice (2 coins): 4 = 1 + 2 + 1 = 1*0 + 2*1 + 1*2 (0 omitted from now on);
3-sided dice (3 long triangular prisms): 27 = 8 + 12 + 6 + 1 = 12*1 + 6*2 + 1*3;
4-sided dice (4 long square prisms or 4 tetrahedrons): 256 = 81 + 108 + 54 + 12 + 1 = 108*1 + 54*2 + 12*3 + 1*4;
5-sided dice (5 long pentagonal prisms): 3125 = 1024 + 1280 + 640 + 160 + 20 + 1 = 1280*1 + 640*2 + 160*3 + 20*4 + 1*5;
6-sided dice (6 cubes): 46656 = 15625 + 18750 + 9375 + 2500 + 375 + 30 + 1 = 18750*1 + 9375*2 + 2500*3 + 375*4 + 30*5 + 1*6.
(End)
For each n >= 1 there is a graph on a(n) vertices whose largest independent set has size n and whose independent set sequence is constant (specifically, for each k=1,2,...,n, the graph has n^n independent sets of size k). There is no graph of smaller order with this property (Ball et al. 2019). - David Galvin, Jun 13 2019
For n >= 2 and 1 <= k <= n, a(n)*(n + 1)/4 + a(n)*(k - 1)*(n + 1 - k)/2*n is equal to the sum over all words w = w(1)...w(n) of length n over the alphabet {1, 2, ..., n} of the following quantity: Sum_{i=1..w(k)} w(i). Inspired by Problem 12432 in the AMM (see links). - Sela Fried, Dec 10 2023
Also, dimension of the unique cohomology group of the smallest interval containing the poset of partitions decorated by Perm, i.e. the poset of pointed partitions. - Bérénice Delcroix-Oger, Jun 25 2025

			G.f. = 1 + x + 4*x^2 + 27*x^3 + 256*x^4 + 3125*x^5 + 46656*x^6 + 823543*x^7 + ...

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, pp. 62, 63, 87.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 173, #39.
A. P. Prudnikov, Yu. A. Brychkov and O.I. Marichev, "Integrals and Series", Volume 1: "Elementary Functions", Chapter 4: "Finite Sums", New York, Gordon and Breach Science Publishers, 1986-1992, Eq. (4.2.2.37)
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).

Kenny Lau, Table of n, a(n) for n = 0..385 [First 100 terms computed by T. D. Noe]
Taylor Ball, David Galvin, Katie Hyry, and Kyle Weingartner, Independent set and matching permutations, arXiv:1901.06579 [math.CO], 2019.
Arthur T. Benjamin and Fritz Juhnke, Another way of counting n^n, SIAM J. Discrete Math., Vol. 5, No. 3 (1992), pp. 377-379. - _N. J. A. Sloane_, Jun 09 2011
H. Bottomley, Illustration of initial terms.
H. J. Brothers and J. A. Knox, New closed-form approximations to the logarithmic constant e, The Mathematical Intelligencer, Vol. 20 (4), 1998, pp. 25-29. (Sequence appears as formula in Eq. (8))
C. Chauve, S. Dulucq and O. Guibert, Enumeration of some labeled trees, Proceedings of FPSAC/SFCA 2000 (Moscow), Springer, pp. 146-157.
Bérénice Delcroix-Oger and Clément Dupont, Lie-operads and operadic modules from poset cohomology, arXiv:2505.06094 [math.CO], 2025. See p. 34.
Frank Ellermann, Illustration of binomial transforms.
José María Grau and Antonio M. Oller-Marcén, On the last digit and the last non-zero digit of n^n in base b, Bulletin of the Korean Mathematical Society, Vol. 51, No. 5 (2014), pp. 1325-1337; arXiv preprint, arXiv:1203.4066 [math.NT], 2012.
Nick Hobson, Solution to puzzle 48: Exponential equation.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 36.
Steven J. Miller (ed.), Exercises to "The Theory and Applications of Benford's Law", Princeton University Press, 2015.
Mustafa Obaid et al., The number of complete exceptional sequences for a Dynkin algebra, arXiv preprint arXiv:1307.7573 [math.RT], 2013.
Franck Ramaharo, A generating polynomial for the pretzel knot, arXiv:1805.10680 [math.CO], 2018.
E. Vigren (Proposer), Problem 12432, Amer. Math. Monthly 130 (2023), p. 953.
Elena L. Wang and Guoce Xin, On Ward Numbers and Increasing Schröder Trees, arXiv:2507.15654 [math.CO], 2025. See pp. 12-13.
Eric Weisstein's World of Mathematics, Hadamard's Maximum Determinant Problem.
Eric Weisstein's World of Mathematics, Hankel Matrix.
Dimitri Zvonkine, An algebra of power series..., arXiv:math/0403092 [math.AG], 2004.
Index entries for "core" sequences
Index entries for sequences related to rooted trees
Index entries for sequences related to Benford's law

Cf. A000107, A000169, A000272, A001372, A007778, A007830, A008785-A008791, A019538, A048993, A008279, A085741, A062206, A212333.
First column of triangle A055858. Row sums of A066324.
Cf. A001923 (partial sums), A002109 (partial products), A007781 (first differences), A066588 (sum of digits).
Cf. A056665, A081721, A130293, A168658, A275549-A275558 (various classes of endofunctions).
Cf. A174824, A204688.
Cf. A055137, A083648.

a000312 n = n ^ n
a000312_list = zipWith (^) [0..] [0..]  -- Reinhard Zumkeller, Jul 07 2012

A000312 := n->n^n: seq(A000312(n), n=0..17);

Array[ #^# &, 16] (* Vladimir Joseph Stephan Orlovsky, May 01 2008 *)
Table[Sum[StirlingS2[n, i] i! Binomial[n, i], {i, 0, n}], {n, 0, 20}] (* Geoffrey Critzer, Mar 17 2009 *)
a[ n_] := If[ n < 1, Boole[n == 0], n^n]; (* Michael Somos, May 24 2014 *)
a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ 1 / (1 + LambertW[-x]), {x, 0, n}]]; (* Michael Somos, May 24 2014 *)
a[ n_] := If[n < 0, 0, n! SeriesCoefficient[ Nest[ 1 / (1 - x / (1 - Integrate[#, x])) &, 1 + O[x], n], {x, 0, n}]]; (* Michael Somos, May 24 2014 *)
a[ n_] := If[ n < 0, 0, With[{m = n + 1}, m! SeriesCoefficient[ InverseSeries[ Series[ (x - 1) Log[1 - x], {x, 0, m}]], m]]]; (* Michael Somos, May 24 2014 *)

A000312[n]:=if n=0 then 1 else n^n$
makelist(A000312[n],n,0,30); /* Martin Ettl, Oct 29 2012 */

PARI
```
{a(n) = n^n};
```

is(n)=my(b,k=ispower(n,,&b));if(k,for(e=1,valuation(k,b), if(k/b^e == e, return(1)))); n==1 \\ Charles R Greathouse IV, Jan 14 2013

{a(n) = my(A = 1 + O(x)); if( n<0, 0, for(k=1, n, A = 1 / (1 - x / (1 - intformal( A)))); n! * polcoeff( A, n))}; /* Michael Somos, May 24 2014 */

def A000312(n): return n**n # Chai Wah Wu, Nov 07 2022

a(n-1) = -Sum_{i=1..n} (-1)^i*i*n^(n-1-i)*binomial(n, i). - Yong Kong (ykong(AT)curagen.com), Dec 28 2000
E.g.f.: 1/(1 + W(-x)), W(x) = principal branch of Lambert's function.
a(n) = Sum_{k>=0} binomial(n, k)*Stirling2(n, k)*k! = Sum_{k>=0} A008279(n,k)*A048993(n,k) = Sum_{k>=0} A019538(n,k)*A007318(n,k). - Philippe Deléham, Dec 14 2003
E.g.f.: 1/(1 - T), where T = T(x) is Euler's tree function (see A000169).
a(n) = A000169(n+1)*A128433(n+1,1)/A128434(n+1,1). - Reinhard Zumkeller, Mar 03 2007
Comment on power series with denominators a(n): Let f(x) = 1 + Sum_{n>=1} x^n/n^n. Then as x -> infinity, f(x) ~ exp(x/e)*sqrt(2*Pi*x/e). - Philippe Flajolet, Sep 11 2008
E.g.f.: 1 - exp(W(-x)) with an offset of 1 where W(x) = principal branch of Lambert's function. - Vladimir Kruchinin, Sep 15 2010
a(n) = (n-1)*a(n-1) + Sum_{i=1..n} binomial(n, i)*a(i-1)*a(n-i). - Vladimir Shevelev, Sep 30 2010
With an offset of 1, the e.g.f. is the compositional inverse ((x - 1)*log(1 - x))^(-1) = x + x^2/2! + 4*x^3/3! + 27*x^4/4! + .... - Peter Bala, Dec 09 2011
a(n) = denominator((1 + 1/n)^n) for n > 0. - Jean-François Alcover, Jan 14 2013
a(n) = A089072(n,n) for n > 0. - Reinhard Zumkeller, Mar 18 2013
a(n) = (n-1)^(n-1)*(2*n) + Sum_{i=1..n-2} binomial(n, i)*(i^i*(n-i-1)^(n-i-1)), n > 1, a(0) = 1, a(1) = 1. - Vladimir Kruchinin, Nov 28 2014
log(a(n)) = lim_{k->infinity} k*(n^(1+1/k) - n). - Richard R. Forberg, Feb 04 2015
From Ilya Gutkovskiy, Jun 18 2016: (Start)
Sum_{n>=1} 1/a(n) = 1.291285997... = A073009.
Sum_{n>=1} 1/a(n)^2 = 1.063887103... = A086648.
Sum_{n>=1} n!/a(n) = 1.879853862... = A094082. (End)
A000169(n+1)/a(n) -> e, as n -> oo. - Daniel Suteu, Jul 23 2016
a(n) = n!*Product_{k=1..n} binomial(n, k)/Product_{k=1..n-1} binomial(n-1, k) = n!*A001142(n)/A001142(n-1). - Tony Foster III, Sep 05 2018
a(n-1) = abs(p_n(2-n)), for n > 2, the single local extremum of the n-th row polynomial of A055137 with Bagula's sign convention. - Tom Copeland, Nov 15 2019
Sum_{n>=1} (-1)^(n+1)/a(n) = A083648. - Amiram Eldar, Jun 25 2021
Limit_{n->oo} (a(n+1)/a(n) - a(n)/a(n-1)) = e (see Brothers/Knox link). - Harlan J. Brothers, Oct 24 2021
Conjecture: a(n) = Sum_{i=0..n} A048994(n, i) * A048993(n+i, n) for n >= 0; proved by Mike Earnest, see link at A354797. - Werner Schulte, Jun 19 2022

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

Original entry on oeis.org

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

A090878 Numerator of Integral_{x=0..infinity} exp(-x)*(1+x/n)^n dx.

Original entry on oeis.org

2, 5, 26, 103, 2194, 1223, 472730, 556403, 21323986, 7281587, 125858034202, 180451625, 121437725363954, 595953719897, 26649932810926, 3211211914492699, 285050975993898158530, 549689343118061, 640611888918574971191834

Offset: 1

Robert G. Wilson v, Feb 13 2004

Also numerators of e_n(n) where e_n(x) is the exponential sum function exp_n(x) and where denominators are given by either A095996 (largest divisor of n! that is coprime to n) or A036503 (denominator of n^(n-2)/n!). - Gerald McGarvey, Nov 14 2005
a(n) is a multiple of A120266(n) or equals A120266(n), A120266(n) is numerator of Sum_{k=0..n} n^k/k!, the integral = (n-1)!/n^(n-1) * the Sum. - Gerald McGarvey, Apr 17 2008
The integral = (1/n^n)*A063170[n] (Schenker sums with n-th term, Integral_{x>0} exp(-x)*(n+x)^n dx). - Gerald McGarvey, Apr 17 2008
Expected value in the birthday paradox problem.  Let X be a random variable that assigns to each f:{1,2,...,n+1}->{1,2,...,n} the smallest k in {2,3,...,n+1} such that f(k)=f(j) for some j < k.  a(n)/A036505(offset=1) = E(X) the expected value of X.  For n=365 E(X) is (surprising low) approximately 24. - Geoffrey Critzer, May 18 2013
Also numerator of Sum_{k=0..n} binomial(n,k)*(k/n)^k*((n-k)/n)^(n-k) [Prodinger]. N. J. A. Sloane, Jul 31 2013

G. C. Greubel, Table of n, a(n) for n = 1..250
Helmut Prodinger, An identity conjectured by Lacasse via the tree function, Electronic Journal of Combinatorics, 20(3) (2013), #P7.
Eric Weisstein, Exponential Sum Function

Denominators are in A036505.

Cf. A120266, A063170.

[Numerator((&+[Binomial(n,k)*(k/n)^k*((n-k)/n)^(n-k): k in [0..n]])): n in [1..20]]; // G. C. Greubel, Feb 08 2019

f[n_]:= Integrate[E^(-x)*(1+x/n)^n, {x,0,Infinity}]; Table[Numerator[ f[n]], {n, 1, 20}]
Table[Numerator[1 + Sum[If[k==0,1,Binomial[n,k]*(k/n)^k*((n-k)/n)^(n-k)], {k,0,n-1}]], {n,1,20}] (* G. C. Greubel, Feb 08 2019 *)

vector(20, n, numerator(sum(k=0, n, binomial(n,k)*(k/n)^k*((n-k)/n)^(n-k)))) \\ G. C. Greubel, Feb 08 2019

[numerator(sum(binomial(n,k)*(k/n)^k*((n-k)/n)^(n-k) for k in (0..n))) for n in (1..20)] # G. C. Greubel, Feb 08 2019

a(n) = A036505(n-1)*Sum_{k=0..n} (A128433(n)/A128434(n)). - Reinhard Zumkeller, Mar 03 2007

Definition corrected by Gerald McGarvey, Apr 17 2008

A036505 Numerator of (n+1)^n/n!.

Original entry on oeis.org

1, 2, 9, 32, 625, 324, 117649, 131072, 4782969, 1562500, 25937424601, 35831808, 23298085122481, 110730297608, 4805419921875, 562949953421312, 48661191875666868481, 91507169819844, 104127350297911241532841, 640000000000000000, 865405750887126927009

Offset: 0

N. J. A. Sloane

Also denominator of Sum_{k=0..n} binomial(n,k)*(k/n)^k*((n-k)/n)^(n-k) [Prodinger]. - N. J. A. Sloane, Jul 31 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Helmut Prodinger, An identity conjectured by Lacasse via the tree function, Electronic Journal of Combinatorics, 20(3) (2013), #P7.

Cf. A036503, A063170.

Cf. A095996 (denominators).

List([0..20], n -> NumeratorRat((n+1)^n/Factorial(n))); # Muniru A Asiru, Feb 12 2018

[Numerator((n+1)^n/Factorial(n)): n in [0..20]]; // Vincenzo Librandi, Sep 10 2013

a:=n -> numer((n+1)^n/factorial(n)):  A036505 := [seq(a(n), n=0..20)]; # Muniru A Asiru, Feb 12 2018

CoefficientList[Series[1/(1 + ProductLog[-x]), {x, 0, 21}], x] // Numerator // Rest (* Jean-François Alcover, Feb 04 2013, after Vladimir Kruchinin *)

my(x='x+O('x^30)); apply(x -> numerator(x), Vec(-1+1/(1+lambertw(-x)))) \\ G. C. Greubel and Michel Marcus, Feb 08 2019

[numerator((n+1)^n/factorial(n)) for n in (0..20)] # G. C. Greubel, Feb 08 2019

a(n) = A090878(n+1)/Sum_{k=0..n+1} (A128433(n+1)/A128434(n+1)). - Reinhard Zumkeller, Mar 03 2007
G.f.: -x*e^(-LambertW(-x))/((LambertW(-x)+1)*LambertW(-x)). - Vladimir Kruchinin, Feb 04 2013
A simpler g.f. is 1/(1 + LambertW(-x)). - Jean-François Alcover, Feb 04 2013

A319861 Triangle read by rows, 0 <= k <= n: T(n,k) is the numerator of the k-th Bernstein basis polynomial of degree n evaluated at the interval midpoint t = 1/2; denominator is A319862.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 3, 1, 1, 1, 5, 5, 5, 5, 1, 1, 3, 15, 5, 15, 3, 1, 1, 7, 21, 35, 35, 21, 7, 1, 1, 1, 7, 7, 35, 7, 7, 1, 1, 1, 9, 9, 21, 63, 63, 21, 9, 9, 1, 1, 5, 45, 15, 105, 63, 105, 15, 45, 5, 1, 1, 11, 55, 165, 165, 231, 231, 165, 165, 55, 11, 1

Offset: 0

Franck Maminirina Ramaharo, Sep 29 2018

In Computer-Aided Geometric Design, the affine combination Sum_{k=0..n} (T(n,k)/A319862(n,k))*P_k is the halfway point for the Bézier curve of degree n defined by the control points P_k, k = 0, 1, ..., n.

			Triangle begins:
  1;
  1, 1;
  1, 1,  1;
  1, 3,  3,  1;
  1, 1,  3,  1,   1;
  1, 5,  5,  5,   5,  1;
  1, 3, 15,  5,  15,  3,   1;
  1, 7, 21, 35,  35, 21,   7,  1;
  1, 1,  7,  7,  35,  7,   7,  1,  1;
  1, 9,  9, 21,  63, 63,  21,  9,  9, 1;
  1, 5, 45, 15, 105, 63, 105, 15, 45, 5, 1;
  ...

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
American Mathematical Society, From Bézier to Bernstein
Rita T. Farouki, The Bernstein polynomial basis: A centennial retrospective, Computer Aided Geometric Design Vol. 29 (2012), 379-419.
Ron Goldman, Pyramid Algorithms. A Dynamic Programming Approach to Curves and Surfaces for Geometric Modeling, Morgan Kaufmann Publishers, 2002, Chap. 5.
Eric Weisstein's World of Mathematics, Bernstein Polynomial
Wikipedia, Bernstein polynomial

Cf. A007318, A082907, A128433, A128434, A319862.

Flat(List([0..11],n->List([0..n],k->Binomial(n,k)/Gcd(Binomial(n,k),2^n)))); # Muniru A Asiru, Sep 30 2018

a:=(n,k)->binomial(n,k)/gcd(binomial(n,k),2^n): seq(seq(a(n,k),k=0..n),n=0..11); # Muniru A Asiru, Sep 30 2018

T[n_, k_] = Binomial[n, k]/GCD[Binomial[n, k], 2^n];
tabl[nn_] = TableForm[Table[T[n, k], {n, 0, nn}, {k, 0, n}]];

T(n,k) := binomial(n, k)/gcd(binomial(n, k), 2^n)$
tabl(nn) := for n:0 thru nn do print(makelist(T(n, k), k, 0, n))$

flatten([[numerator(binomial(n,k)/2^n) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jul 19 2021

T(n, k) = numerator of binomial(n,k)/2^n.
T(n, k) = binomial(n,k)/A082907(n,k).
T(n, k)/A319862(n,k) = binomial(n,k)/2^n.
T(n, n-k) = T(n,k).
T(n, 0) = 1.
T(n, 1) = A000265(n) (with offset 0, following Peter Luschny's formula).
T(n, 2) = A069834(n-1), n > 1.
Sum_{k=0..n} 2*k*T(n,k)/A319862(n,k) = n.
Sum_{k=0..n} 2*k^2*T(n,k)/A319862(n,k) = A000217(n).

A319862 Triangle read by rows, 0 <= k <= n: T(n,k) is the denominator of the k-th Bernstein basis polynomial of degree n evaluated at the interval midpoint t = 1/2; numerator is A319861.

Original entry on oeis.org

1, 2, 2, 4, 2, 4, 8, 8, 8, 8, 16, 4, 8, 4, 16, 32, 32, 16, 16, 32, 32, 64, 32, 64, 16, 64, 32, 64, 128, 128, 128, 128, 128, 128, 128, 128, 256, 32, 64, 32, 128, 32, 64, 32, 256, 512, 512, 128, 128, 256, 256, 128, 128, 512, 512, 1024, 512, 1024, 128, 512, 256, 512, 128, 1024, 512, 1024

Offset: 0

Franck Maminirina Ramaharo, Sep 29 2018

			Triangle begins:
    1;
    2,   2;
    4,   2,   4;
    8,   8,   8,   8;
   16,   4,   8,   4,  16;
   32,  32,  16,  16,  32,  32;
   64,  32,  64,  16,  64,  32,  64;
  128, 128, 128, 128, 128, 128, 128, 128;
  256,  32,  64,  32, 128,  32,  64,  32, 256;
  512, 512, 128, 128, 256, 256, 128, 128, 512, 512;
  ...

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
American Mathematical Society, From Bézier to Bernstein
Rita T. Farouki, The Bernstein polynomial basis: A centennial retrospective, Computer Aided Geometric Design Vol. 29 (2012), 379-419.
Ron Goldman, Pyramid Algorithms. A Dynamic Programming Approach to Curves and Surfaces for Geometric Modeling, Morgan Kaufmann Publishers, 2002, Chap. 5.
Eric Weisstein's World of Mathematics, Bernstein Polynomial
Wikipedia, Bernstein polynomial

Cf. A007318, A082907, A128433, A128434, A319861.

Flat(List([0..11],n->List([0..n],k->2^n/Gcd(Binomial(n,k),2^n)))); # Muniru A Asiru, Sep 30 2018

a:=(n,k)->2^n/gcd(binomial(n,k),2^n): seq(seq(a(n,k),k=0..n),n=0..11); # Muniru A Asiru, Sep 30 2018

T[n_, k_] = 2^n/GCD[Binomial[n, k], 2^n];
tabl[nn_] = TableForm[Table[T[n, k], {n, 0, nn}, {k, 0, n}]];

T(n, k) := 2^n/gcd(binomial(n, k), 2^n)$
tabl(nn) := for n:0 thru nn do print(makelist(T(n, k), k, 0, n))$

def A319862(n,k): return denominator(binomial(n,k)/2^n)
flatten([[A319862(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jul 20 2021

T(n, k) = denominator of binomial(n,k)/2^n.
T(n, k) = 2^n/A082907(n,k).
A319862(n, k)/T(n, k) = binomial(n,k)/2^n.
T(n, n-k) = T(n, k).
T(n, 0) = 2^n.
T(n, 1) = A075101(n).

A141692 Triangle read by rows: T(n,k) = n*(binomial(n - 1, k - 1) - binomial(n - 1, k)), 0 <= k <= n.

Original entry on oeis.org

0, -1, 1, -2, 0, 2, -3, -3, 3, 3, -4, -8, 0, 8, 4, -5, -15, -10, 10, 15, 5, -6, -24, -30, 0, 30, 24, 6, -7, -35, -63, -35, 35, 63, 35, 7, -8, -48, -112, -112, 0, 112, 112, 48, 8, -9, -63, -180, -252, -126, 126, 252, 180, 63, 9, -10, -80, -270, -480, -420, 0, 420, 480, 270, 80, 10

Offset: 0

Roger L. Bagula, Sep 09 2008

The row sums are zero.

Row n consists of the coefficients in the expansion of n*(x - 1)*(x + 1)^(n - 1). - Franck Maminirina Ramaharo, Oct 02 2018

			Triangle begins:
    0;
   -1,   1;
   -2,   0,    2;
   -3,  -3,    3,    3;
   -4,  -8,    0,    8,    4;
   -5, -15,  -10,   10,   15,   5;
   -6, -24,  -30,    0,   30,  24,   6;
   -7, -35,  -63,  -35,   35,  63,  35,   7;
   -8, -48, -112, -112,    0, 112, 112,  48,   8;
   -9, -63, -180, -252, -126, 126, 252, 180,  63,  9;
  -10, -80, -270, -480, -420,   0, 420, 480, 270, 80, 10;
  ...

G. C. Greubel, Table of n, a(n) for n = 0..5150 (Rows n=1..100 of triangle, flattened; offset adapted by _Georg Fischer_, Jan 31 2019)
Rida T. Farouki, The Bernstein polynomial basis: A centennial retrospective, Computer Aided Geometric Design Vol. 29 (2012), 379-419.
Eric Weisstein's World of Mathematics, Bernstein Polynomial
Wikipedia, Bernstein polynomial

Cf. A007318, A112467, A128433, A128434.

a:=proc(n,k) n*(binomial(n-1,k-1)-binomial(n-1,k)); end proc: seq(seq(a(n,k),k=0..n),n=0..10); # Muniru A Asiru, Oct 03 2018

Table[Table[n*(Binomial[n - 1, k - 1] - Binomial[n - 1, k]),{k, 0, n}],{n, 0, 12}]//Flatten

T(n, k) := n*(binomial(n - 1, k - 1) - binomial(n - 1, k))$
tabl(nn) := for n:0 thru nn do print(makelist(T(n, k), k, 0, n))$ /* Franck Maminirina Ramaharo, Oct 02 2018 */

T(n,k) = n*(B(1/2;n-1,k-1) - B(1/2;n-1,k))*2^(n - 1), where B(t;n,k) = binomial(n,k)*t^k*(1 - t)^(n - k) denotes the k-th Benstein basis polynomial of degree n.
T(n,k) = n*A112467(n,k).
From Franck Maminirina Ramaharo, Oct 02 2018: (Start)
T(n,k) = -T(n,n-k)
T(n,0) = -n.
T(n,1) = -A067998(n)
E.g.f.: (x*y - y)/(x*y + y - 1)^2.
Sum_{k=0..n} abs(T(n,k)) = 2*A100071(n).
Sum_{k=0..n} T(n,k)^2 = 2*A037965(n).
Sum_{k=0..n} k*T(n,k) = A001787(n).
Sum_{k=0..n} k^2*T(n,k) = A014477(n-1). (End)

Edited, new name and offset corrected by Franck Maminirina Ramaharo, Oct 02 2018

A320085 Triangle read by rows, 0 <= k <= n: T(n,k) is the numerator of the derivative of the k-th Bernstein basis polynomial of degree n evaluated at the interval midpoint t = 1/2; denominator is A320086.

Original entry on oeis.org

0, -1, 1, -1, 0, 1, -3, -3, 3, 3, -1, -1, 0, 1, 1, -5, -15, -5, 5, 15, 5, -3, -3, -15, 0, 15, 3, 3, -7, -35, -63, -35, 35, 63, 35, 7, -1, -3, -7, -7, 0, 7, 7, 3, 1, -9, -63, -45, -63, -63, 63, 63, 45, 63, 9, -5, -5, -135, -15, -105, 0, 105, 15, 135, 5, 5

Offset: 0

Franck Maminirina Ramaharo, Oct 05 2018

If n = 2*k, then T(n,k) = 0 since the k-th Bernstein basis polynomial of degree n has a single unique local maximum occurring at t = k/n, which coincides with the interval midpoint t = 1/2 (T(0,0) = 0 because the only 0 degree Bernstein basis polynomial is the constant 1).

			Triangle begins:
   0;
  -1,   1;
  -1,   0,    1;
  -3,  -3,    3,   3;
  -1,  -1,    0,   1,    1;
  -5, -15,   -5,   5,   15,  5;
  -3,  -3,  -15,   0,   15,  3,   3;
  -7, -35,  -63, -35,   35, 63,  35,  7;
  -1,  -3,   -7,  -7,    0,  7,   7,  3,   1;
  -9, -63,  -45, -63,  -63, 63,  63, 45,  63, 9;
  -5,  -5, -135, -15, -105,  0, 105, 15, 135, 5, 5;
  ...

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Rita T. Farouki, The Bernstein polynomial basis: A centennial retrospective, Computer Aided Geometric Design Vol. 29 (2012), 379-419.
Ron Goldman, Pyramid Algorithms. A Dynamic Programming Approach to Curves and Surfaces for Geometric Modeling, Morgan Kaufmann Publishers, 2002, Chap. 5.
Eric Weisstein's World of Mathematics, Bernstein Polynomial
Wikipedia, Bernstein polynomial

Inspired by A141692.

Cf. A007318, A128433, A128434, A319861, A319862, A320086.

T:=proc(n,k) n*(binomial(n-1,k-1)-binomial(n-1,k))/gcd(n*(binomial(n-1,k-1)-binomial(n-1,k)),2^(n-1)); end proc: seq(seq(T(n,k),k=0..n),n=0..11); # Muniru A Asiru, Oct 06 2018

Table[Numerator[n*(Binomial[n-1, k-1] - Binomial[n-1, k])/2^(n-1)], {n, 0, 12}, {k, 0, n}]//Flatten

T(n, k) := n*(binomial(n - 1, k - 1) - binomial(n - 1, k))/gcd(n*(binomial(n - 1, k - 1) - binomial(n - 1, k)), 2^(n - 1))$
tabl(nn) := for n:0 thru nn do print(makelist(T(n, k), k, 0, n))$

def A320085(n,k): return numerator(n*(binomial(n-1, k-1) - binomial(n-1, k))/2^(n-1))
flatten([[A320085(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jul 19 2021

T(n, k) = numerator of 2*A141692(n,k)/A000079(n).
T(n, k) = n*(binomial(n-1, k-1) - binomial(n-1, k))/gcd(n*(binomial(n-1, k-1) - binomial(n-1, k)), 2^(n-1)).
T(n, n-k) = -T(n,k).
T(n, 0) = -n.
T(2*n+1, 1) = -A000466(n).
T(2*n, 1) = -A069834(n-1), n > 1.
T(n, k)/A320086(n,k) = 4*n*(k/n - 1/2)*A319861(n,k)/A319861(n,k).
Sum_{k=0..n} k*T(n,k)/A320086(n,k) = n.
Sum_{k=0..n} k^2*T(n,k)/A320086(n,k) = n^2.
Sum_{k=0..n} k*(k-1)*T(n,k)/A320086(n,k) = n*(n - 1).

A320086 Triangle read by rows, 0 <= k <= n: T(n,k) is the denominator of the derivative of the k-th Bernstein basis polynomial of degree n evaluated at the interval midpoint t = 1/2; numerator is A320085.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 4, 4, 4, 4, 2, 1, 1, 1, 2, 16, 16, 8, 8, 16, 16, 16, 4, 16, 1, 16, 4, 16, 64, 64, 64, 64, 64, 64, 64, 64, 16, 8, 8, 8, 1, 8, 8, 8, 16, 256, 256, 64, 64, 128, 128, 64, 64, 256, 256, 256, 32, 256, 16, 128, 1, 128, 16, 256, 32, 256

Offset: 0

Franck Maminirina Ramaharo, Oct 05 2018

			Triangle begins:
    1;
    1,   1;
    1,   1,   1;
    4,   4,   4,  4;
    2,   1,   1,  1,   2;
   16,  16,   8,  8,  16,  16;
   16,   4,  16,  1,  16,   4,  16;
   64,  64,  64, 64,  64,  64,  64, 64;
   16,   8,   8,  8,   1,   8,   8,  8,  16;
  256, 256,  64, 64, 128, 128,  64, 64, 256, 256;
  256,  32, 256, 16, 128,   1, 128, 16, 256,  32, 256;
  ...

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Rita T. Farouki, The Bernstein polynomial basis: A centennial retrospective, Computer Aided Geometric Design Vol. 29 (2012), 379-419.
Ron Goldman, Pyramid Algorithms. A Dynamic Programming Approach to Curves and Surfaces for Geometric Modeling, Morgan Kaufmann Publishers, 2002, Chap. 5.
Eric Weisstein's World of Mathematics, Bernstein Polynomial
Wikipedia, Bernstein polynomial

Inspired by A141692.

Cf. A007318, A128433, A128434, A319861, A319862, A320085.

T:=proc(n,k) 2^(n-1)/gcd(n*(binomial(n-1,k-1)-binomial(n-1,k)),2^(n-1)); end proc: seq(seq(T(n,k),k=0..n),n=1..11); # Muniru A Asiru, Oct 06 2018

Table[Denominator[n*(Binomial[n-1, k-1] - Binomial[n-1, k])/2^(n-1)], {n, 0, 12}, {k, 0, n}]//Flatten

T(n, k) := 2^(n - 1)/gcd(n*(binomial(n - 1, k - 1) - binomial(n - 1, k)), 2^(n - 1))$
tabl(nn) := for n:0 thru nn do print(makelist(T(n, k), k, 0, n))$

def A320086(n,k): return denominator(n*(binomial(n-1, k-1) - binomial(n-1, k))/2^(n-1))
flatten([[A320086(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jul 19 2021

T(n, k) = denominator of 2*A141692(n,k)/A000079(n).
T(n, k) = 2^(n-1)/gcd(n*(binomial(n-1, k-1) - binomial(n-1, k)), 2^(n-1)).
T(n, n-k) = T(n,k).
T(n, 0) = A084623(n), n > 0.
T(2*n+1, 1) = A000302(n).

cp's OEIS Frontend

A128434 Triangle, read by rows, T(n,k) = denominator of the maximum of the k-th Bernstein polynomial of degree n; numerator is A128433.

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A000312 a(n) = n^n; number of labeled mappings from n points to themselves (endofunctions).

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Maxima

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

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A090878 Numerator of Integral_{x=0..infinity} exp(-x)*(1+x/n)^n dx.

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A036505 Numerator of (n+1)^n/n!.

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A319861 Triangle read by rows, 0 <= k <= n: T(n,k) is the numerator of the k-th Bernstein basis polynomial of degree n evaluated at the interval midpoint t = 1/2; denominator is A319862.