A055858 -id:A055858 - cp's OEIS frontend

A055862 Fourth column of triangle A055858.

0, 0, 0, 36, 400, 5400, 86436, 1605632, 34012224, 810000000, 21435888100, 624333422592, 19851622826256, 684376513673216, 25430282226562500, 1013309916158361600, 43104723599206637824, 1949779138161985486848

Offset: 0

Wolfdieter Lang Jun 20 2000

a(n)=A055858(n, 3). Cf. A000312, A000169, A055860, A055861.

a(i)=0, i=0, 1, 2; a(n)= (n^2)*(n+1)^(n-2), n >= 3.

E.g.f.: -x*(1+4*x)/2 + (W(-x)^2)*(1-W(-x))/((1+W(-x))*x*2), W(x) principal branch of Lambert's function.

A055863 Fifth column of triangle A055858.

Original entry on oeis.org

0, 0, 0, 0, 320, 4500, 74088, 1404928, 30233088, 729000000, 19487171000, 572305637376, 18324574916544, 635492476982272, 23734930078125000, 949978046398464000, 40569151622782717952, 1841458074930764070912

Offset: 0

Wolfdieter Lang Jun 20 2000

a(n)=A055858(n, 4). Cf. A000312, A000169, A055860-A055862.

a(i)=0, i=0..3; a(n)= (n^3)*(n+1)^(n-3), n >= 4.

E.g.f.: -x*(3+16*x+54*x^2)/12+(W(-x)^2)*(3-7*W(-x)+2*W(-x)^2)/ (12*(1+W(-x))*x), W(x) principal branch of Lambert's function.

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

Original entry on oeis.org

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

A045531 Number of sticky functions: endofunctions of [n] having a fixed point.

Original entry on oeis.org

1, 3, 19, 175, 2101, 31031, 543607, 11012415, 253202761, 6513215599, 185311670611, 5777672071535, 195881901213181, 7174630439858727, 282325794823047151, 11878335717996660991, 532092356706983938321, 25283323623228812584415, 1270184310304975912766347

Offset: 1

Len Smiley

a(n) is also the number of functions f{1,2,...,n}->{1,2,...,n} with at least one element mapped to 1. - Geoffrey Critzer, Dec 10 2012
Equivalently, a(n) is the number of endofunctions with minimum 1. - Olivier Gérard, Aug 02 2016
Number of bargraphs of width n and height n. Equivalently: number of ordered n-tuples of positive integers such that the largest is n. Example: a(3) = 19 because we have 113, 123, 213, 223, 131, 132, 231, 232, 311, 312, 321, 322, 331, 332, 313, 323, 133, 233, and 333. - Emeric Deutsch, Jan 30 2017

Vincenzo Librandi, Table of n, a(n) for n = 1..100
A. Blecher, C. Brennan, A. Knopfmacher and H. Prodinger, The height and width of bargraphs, Discrete Applied Math. 180, (2015), 36-44.

Column |a(n, 2)| of A039621. Row sums of triangle A055858.
Cf. A000312, A066274, A066275, A047969.
Column k=1 of A246049.
Cf. A068996, A350454.

[n^n-(n-1)^n: n in [1..20] ]; // Vincenzo Librandi, May 07 2011

Table[Sum[Binomial[n, i] (n - 1)^(n - i), {i, 1, n}], {n, 1, 20}]

a(n) = sum(k!*binomial(n-1,k-1)*stirling2(n,k),k,1,n); /* Vladimir Kruchinin, Mar 01 2014 */

a(n)=n^n-(n-1)^n; \\ Charles R Greathouse IV, May 08 2011

a(n) = n^n - (n-1)^n.
E.g.f.: (T - x)/(T-T^2), where T=T(x) is Euler's tree function (see A000169).
With interpolated zeros, ceiling(n/2)^ceiling(n/2) - floor(n/2)^ceiling(n/2). - Paul Barry, Jul 13 2005
a(n) = A047969(n,n). - Alford Arnold, May 07 2005
a(n) = Sum_{i=1..n} binomial(n,i)*(i-1)^(i-1)*(n-i)^(n-i) = Sum_{i=1..n} binomial(n,i)*A000312(i-1)*A000312(n-i). - Vladimir Shevelev, Sep 30 2010
a(n) = Sum_{k=1..n} k!*binomial(n-1,k-1)*Stirling2(n,k). - Vladimir Kruchinin, Mar 01 2014
a(n) = A350454(n+1, 1) / (n+1). - Mélika Tebni, Dec 20 2022
Limit_{n->oo} a(n)/n^n = 1 - 1/e = A068996. - Luc Rousseau, Jan 20 2023

A055869 a(n) = (n+1)^n - n^n.

Original entry on oeis.org

1, 5, 37, 369, 4651, 70993, 1273609, 26269505, 612579511, 15937424601, 457696700077, 14381984674225, 490839666661891, 18080919199832609, 715027614225987601, 30214447801957316865, 1358671297852359767791, 64780942222614703957417, 3264460344339686410876021

Offset: 1

Wolfdieter Lang, Jun 20 2000

Number of functions f:[n]->[n+1] such that some x in [n] maps to n+1.

Number of switching generators for a power polyadic n-context ({1..k}, ..., {1..k}, <>) with n=k [Theorems 5 and 6, page 81, in Ignatov]. - Dmitry I. Ignatov, Nov 23 2022

G. C. Greubel, Table of n, a(n) for n = 1..385
D. I. Ignatov, On closure operators related to maximal tricliques in tripartite hypergraphs, Discrete Appl. Math., 249 (2018), 74-84.
D. I. Ignatov, Supporting iPython code for enumeration of switching generators of power polyadic n-contexts for n=k<6, GitHub repository.

Row sums of triangle A055864.

Cf. A055864, A055858, A045531.

[(n+1)^n - n^n: n in [1..40]]; // Vincenzo Librandi, Jan 11 2015

Table[(n+1)^n-n^n,{n,40}] (* Vladimir Joseph Stephan Orlovsky, Nov 20 2009 *)

vector(20, n, (n+1)^n - n^n) \\ Michel Marcus, Jan 10 2015

E.g.f.: W(-x)*(x-1)/((1+W(-x))*x), W(x) principal branch of Lambert's function.
a(n) = Sum_{m=1..n} A055864(n, m).
a(n) = Sum_{i=0..n-1} n^i*C(n, i). - Olivier Gérard, Jun 26 2001
With interpolated zeros, ceiling(n/2)^floor(n/2) - floor(n/2)^floor(n/2). - Paul Barry, Jul 13 2005
a(n) = Sum_{k=1..n} (-1)^(n-k)*k!*Stirling2(n,k)*binomial(n+k-1,n). - Vladimir Kruchinin, Sep 20 2015

More terms from Vincenzo Librandi, Jan 11 2015

A055865 Second column of triangle A055864.

Original entry on oeis.org

0, 2, 12, 100, 1080, 14406, 229376, 4251528, 90000000, 2143588810, 56757583872, 1654301902188, 52644347205632, 1816448730468750, 67553994410557440, 2694045224950414864, 114692890480116793344, 5191945444217181018258, 249036800000000000000000, 12617615847934310595791220

Offset: 1

Wolfdieter Lang Jun 20 2000

Paolo Xausa, Table of n, a(n) for n = 1..350
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets

Cf. A055864, A000272, A055858.

A055865[n_] := If[n == 1, 0, n*(n+1)^(n-2)];
Array[A055865, 25] (* Paolo Xausa, Aug 07 2025 *)

a(1) = 0; a(n) = n*(n+1)^(n-2), n >= 2.
E.g.f.: -1/2*(-W(-x)^2+x^2)/x, W(x) principal branch of Lambert's function.
a(n) = A055864(n, 2).

A055860 a(n) = A000169(n+1) if n > 0; a(0) = 0.

Original entry on oeis.org

0, 2, 9, 64, 625, 7776, 117649, 2097152, 43046721, 1000000000, 25937424601, 743008370688, 23298085122481, 793714773254144, 29192926025390625, 1152921504606846976, 48661191875666868481, 2185911559738696531968

Offset: 0

Wolfdieter Lang, Jun 20 2000

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 67

Second column of triangle A055858. Cf. A000169, A055858, A000312.

a(0) = 0; for n >= 1, a(n) = (n+1)^n.

E.g.f.: -W(-x)/((1+W(-x))*x) - 1 = -(d/dx)W(x) - 1, W(x) principal branch of Lambert's function.

A055864 Coefficient triangle for certain polynomials.

Original entry on oeis.org

1, 3, 2, 16, 12, 9, 125, 100, 80, 64, 1296, 1080, 900, 750, 625, 16807, 14406, 12348, 10584, 9072, 7776, 262144, 229376, 200704, 175616, 153664, 134456, 117649, 4782969, 4251528, 3779136, 3359232, 2985984, 2654208, 2359296, 2097152

Offset: 1

Wolfdieter Lang, Jun 20 2000

The coefficients of the partner polynomials are found in triangle A055858.

			Fourth row polynomial (n=4): p(4,x) = 125+100*x+80*x^2+64*x^3.
Triangle begins:
        1;
        3,       2;
       16,      12,       9;
      125,     100,      80,      64;
     1296,    1080,     900,     750,     625;
    16807,   14406,   12348,   10584,    9072,    7776;
   262144,  229376,  200704,  175616,  153664,  134456,  117649;
  4782969, 4251528, 3779136, 3359232, 2985984, 2654208, 2359296, 2097152;
  ...

Column sequences are: A000272(n+1), n >= 1, A055865, A055070, A055867, A055868 for m=1..5.

Main diagonal gives A000169.

a[n_, m_] /; n < m = 0; a[n_, m_] := n^(m-1)*(n+1)^(n-m); Table[a[n, m], {n, 1, 8}, {m, 1, n}] // Flatten (* Jean-François Alcover, Jun 20 2013 *)

a(n, m)=0 if n= m >= 1;

E.g.f. for column m: A(m, x); A(1, x)=-(W(-x)/x+1); recursion: A(m, x) = A(m-1, x)-int(A(m-1, x), x)/x-(((m-1)^(m-1))/m)* (x^(m-1))/(m-1)!, m >= 2; W(x) principal branch of Lambert's function.

A055867 Fourth column of triangle A055864.

Original entry on oeis.org

0, 0, 0, 64, 750, 10584, 175616, 3359232, 72900000, 1771561000, 47692136448, 1409582685888, 45392319784448, 1582328671875000, 59373627899904000, 2386420683693101056, 102303226385042448384, 4659806991485780193672

Offset: 1

Wolfdieter Lang, Jun 20 2000

a(n)=A055864(n, 4). Cf. A000272, A055865, A055070, A055866, A055858.

A055867:=n->n^3*(n+1)^(n-4): 0,0,0,seq(A055867(n), n=4..30); # Wesley Ivan Hurt, Jan 27 2017

a(i)=0 for i=1, 2, 3; a(n) = n^3*(n+1)^(n-4), n >= 4.

E.g.f.: (9*W(-x)^2-14*W(-x)^3+3*W(-x)^4-9*x^2-32*x^3-81*x^4)/(72*x), W(x) principal branch of Lambert's function.

A055861 Essentially A053506 but with leading 0 (instead of 1) and offset 0.

Original entry on oeis.org

0, 0, 6, 48, 500, 6480, 100842, 1835008, 38263752, 900000000, 23579476910, 681091006464, 21505924728444, 737020860878848, 27246730957031250, 1080863910568919040, 45798768824157052688, 2064472028642102280192

Offset: 0

Wolfdieter Lang, Jun 20 2000

Third column of triangle A055858.

Cf. A053506, A000312, A000169, A055860.

a(0) = 0 = a(1); a(n) = n*(n+1)^(n-1), n >= 2.

E.g.f.: -x + W(-x)^2/((1+W(-x))*x) = ((d/dx)W(-x)^2)/2-x, W(x) principal branch of Lambert's function.

cp's OEIS Frontend

A055862 Fourth column of triangle A055858.

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A055863 Fifth column of triangle A055858.

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

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A045531 Number of sticky functions: endofunctions of [n] having a fixed point.

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A055869 a(n) = (n+1)^n - n^n.

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A055865 Second column of triangle A055864.

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A055860 a(n) = A000169(n+1) if n > 0; a(0) = 0.

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A055864 Coefficient triangle for certain polynomials.

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