A005727 -id:A005727 - cp's OEIS frontend

A005728 Number of fractions in Farey series of order n.

1, 2, 3, 5, 7, 11, 13, 19, 23, 29, 33, 43, 47, 59, 65, 73, 81, 97, 103, 121, 129, 141, 151, 173, 181, 201, 213, 231, 243, 271, 279, 309, 325, 345, 361, 385, 397, 433, 451, 475, 491, 531, 543, 585, 605, 629, 651, 697, 713, 755, 775, 807, 831, 883, 901, 941, 965

Offset: 0

N. J. A. Sloane

Sometimes called Phi(n).
Leo Moser found an interesting way to generate this sequence, see Gardner.
a(n) is a prime number for nine consecutive values of n: n = 1, 2, 3, 4, 5, 6, 7, 8, 9. - Altug Alkan, Sep 26 2015
Named after the English geologist and writer John Farey, Sr. (1766-1826). - Amiram Eldar, Jun 17 2021

			a(5)=11 because the fractions are 0/1, 1/5, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5, 1/1.

Martin Gardner, The Last Recreations, 1997, chapter 12.
Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Mathematics, a foundation for computer science, Chapter 4.5 - Relative Primality, pages 118 - 120 and Chapter 9 - Asymptotics, Problem 6, pages 448 - 449, Addison-Wesley Publishing Co., Reading, Mass., 1989.
William Judson LeVeque, Topics in Number Theory, Addison-Wesley, Reading, MA, 2 vols., 1956, Vol. 1, p. 154.
Andrey O. Matveev, Farey Sequences, De Gruyter, 2017, Table 1.7.
Leo Moser, Solution to Problem P42, Canadian Mathematical Bulletin, Vol. 5, No. 3 (1962), pp. 312-313.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Antoine Mathys, Table of n, a(n) for n = 0..20000 (terms 0 to 1000 from T. D. Noe)
Richard K. Guy, Letter to N. J. A. Sloane, 1986.
Richard K. Guy, Letter to N. J. A. Sloane, 1987.
Richard K. Guy, The strong law of small numbers. Amer. Math. Monthly, Vol. 95, No. 8 (1988), pp. 697-712.
Richard K. Guy, The strong law of small numbers. Amer. Math. Monthly, Vol. 95, No. 8 (1988), pp. 697-712. [Annotated scanned copy]
Brady Haran and Grant Sanderson, Prime Pyramid (with 3Blue1Brown), Numberphile video (2022).
Sameen Ahmed Khan, Mathematica notebook.
Sameen Ahmed Khan, How Many Equivalent Resistances?, RESONANCE, May 2012.
Sameen Ahmed Khan, Farey sequences and resistor networks, Proc. Indian Acad. Sci. (Math. Sci.), Vol. 122, No. 2 (May 2012), pp. 153-162.
Sameen Ahmed Khan, Beginning to count the number of equivalent resistances, Indian Journal of Science and Technology, Vol. 9, No. 44 (2016), pp. 1-7.
Andrey O. Matveev, Farey Sequences: Errata + Haskell code
Shmuel Schreiber and N. J. A. Sloane, Correspondence, 1980.
N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021. (Includes this sequence)
Vladimir Sukhoy and Alexander Stoytchev, Numerical error analysis of the ICZT algorithm for chirp contours on the unit circle, Scientific Reports, Vol. 10, Article No. 4852 (2020).
Vladimir Sukhoy and Alexander Stoytchev, Formulas and algorithms for the length of a Farey sequence, Scientific Reports, Vol. 11 (2021), Article No. 22218.
Eric Weisstein's World of Mathematics, Farey Sequence.
Wikipedia, Farey sequence.

For the Farey series see A006842/A006843.
Essentially the same as A049643.
Cf. A002088, A055197, A055201.

List([0..60],n->Sum([1..n],i->Phi(i)))+1; # Muniru A Asiru, Jul 31 2018

a005728 n = a005728_list
a005728_list = scanl (+) 1 a000010_list
-- Reinhard Zumkeller, Aug 04 2012

[1] cat [n le 1 select 2 else Self(n-1)+EulerPhi(n): n in [1..60]]; // Vincenzo Librandi, Sep 27 2015

A005728 := proc(n)
    1+add(numtheory[phi](i),i=1..n) ;
end proc:
seq(A005728(n),n=0..80) ; # R. J. Mathar, Nov 29 2017

Accumulate@ Array[ EulerPhi, 54, 0] + 1
f[n_] := 1 + Sum[ EulerPhi[m], {m, n}]; Array[f, 55, 0] (* or *)
f[n_] := (Sum[ MoebiusMu[m] Floor[n/m]^2, {m, n}] + 3)/2; f[0] = 1; Array[f, 55, 0] (* or *)
f[n_] := n (n + 3)/2 - Sum[f[Floor[n/m]], {m, 2, n}]; f[0] = 1; Array[f, 55, 0] (* Robert G. Wilson v, Sep 26 2015 *)
a[n_] := If[n == 0, 1, FareySequence[n] // Length];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Jul 16 2022 *)

a(n)=1+sum(k=1,n,eulerphi(k)) \\ Charles R Greathouse IV, Jun 03 2013

from functools import lru_cache
@lru_cache(maxsize=None)
def A005728(n): # based on second formula in A018805
    if n == 0:
        return 1
    c, j = -2, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2-j)*(2*A005728(k1)-3)
        j, k1 = j2, n//j2
    return (n*(n-1)-c+j)//2 # Chai Wah Wu, Mar 24 2021

a(n) = 1 + Sum_{i=1..n} phi(i).
a(n) = n*(n+3)/2 - Sum_{k=2..n} a(floor(n/k)). - David W. Wilson, May 25 2002
a(n) = a(n-1) + phi(n) with a(0) = 1. - Arkadiusz Wesolowski, Oct 13 2012
a(n) = 1 + A002088(n). - Robert G. Wilson v, Sep 26 2015

A000248 Expansion of e.g.f. exp(x*exp(x)).

Original entry on oeis.org

1, 1, 3, 10, 41, 196, 1057, 6322, 41393, 293608, 2237921, 18210094, 157329097, 1436630092, 13810863809, 139305550066, 1469959371233, 16184586405328, 185504221191745, 2208841954063318, 27272621155678841, 348586218389733556, 4605223387997411873

Offset: 0

N. J. A. Sloane, Simon Plouffe

Number of forests with n nodes and height at most 1.
Equivalently, number of idempotent mappings f from a set of n elements into itself (i.e., satisfying f o f = f). - Robert FERREOL, Oct 11 2007
In other words, a(n) = number of idempotents in the full semigroup of maps from [1..n] to itself. [Tainiter]
a(n) is the number of ways to select a set partition of {1,2,...,n} and then designate one element in each block (cell) of the partition.
Let set B have cardinality n. Then a(n) is the number of functions f:D->C over all partitions {D,C} of B. See the example in the Example Section below. We note that f:empty set->B is designated as the null function, whereas f:B->empty set is undefined unless B itself is empty. - Dennis P. Walsh, Dec 05 2013
In physics, a(n) would be interpreted as the number of projection operators P on S_n, i.e., ones satisfying P^2 = P. Example: a particle with a half-integer spin s has a spin space with 2s+1 base states which admits a(2s+1) linear projection operators (including the identity). These are important because they satisfy the operator identity exp(zU) = 1+(exp(z)-1)*U, valid for any complex z. - Stanislav Sykora, Nov 03 2016

			a(3)=10 since, for B={1,2,3}, we have 10 functions: 1 function of the type f:empty set->B; 6 functions of the type f:{x}->B\{x}; and 3 functions of the type f:{x,y}->B\{x,y}. - _Dennis P. Walsh_, Dec 05 2013

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 91.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.32(d).

Alois P. Heinz, Table of n, a(n) for n = 0..541 (first 101 terms from T. D. Noe)
Rudolf Berghammer, Jules Desharnais, and Michael Winter, Counting Specific Classes of Relations Regarding Fixed Points and Reflexive Points, arXiv:2505.00140 [cs.DM], 2025. See p. 24.
Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.
Giulio Cerbai and Anders Claesson, Counting fixed-point-free Cayley permutations, arXiv:2507.09304 [math.CO], 2025. See p. 25.
Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, 2009; see page 131.
Xing Gao and William F. Keigher, Interlacing of Hurwitz series, Communications in Algebra, 45:5 (2017), 2163-2185, DOI: 10.1080/00927872.2016.1226885. See Ex. 2.13.
Bernhard Harris and Lowell Schoenfeld, The number of idempotent elements in symmetric semigroups, J. Combin. Theory, 3 (1967), 122-135.
Bernhard Harris and Lowell Schoenfeld, Asymptotic expansions for the coefficients of analytic functions, Illinois Journal of Mathematics, Volume 12, Issue 2 (1968), 264-277.
Gottfried Helms, Pascalmatrix tetrated
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 117
Vaclav Kotesovec, Graph - the asymptotic ratio (1000 terms)
Nate Kube and Frank Ruskey, Sequences That Satisfy a(n-a(n))=0, Journal of Integer Sequences, Vol. 8 (2005), Article 05.5.5.
John Riordan, Forests of labeled trees, J. Combin. Theory, 5 (1968), 90-103.
John Riordan, Letter to N. J. A. Sloane, Oct. 1970
John Riordan and N. J. A. Sloane, Correspondence, 1974
Emre Sen, Exceptional Sequences and Idempotent Functions, arXiv:1909.05887 [math.RT], 2019.
Melvin Tainiter, A characterization of idempotents in semigroups, J. Combinat. Theory, 5 (1968), 370-373.
Haoliang Wang and Robert Simon, The Analysis of Synchronous All-to-All Communication Protocols for Wireless Systems, Q2SWinet'18: Proceedings of the 14th ACM International Symposium on QoS and Security for Wireless and Mobile Networks (2018), 39-48.

First row of array A098697.
Row sums of A133399.
Column k=1 of A210725, A279636.
Column k=2 of A245501.
Cf. A005727, A048993.

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(x*Exp(x)))); [Factorial(n-1)*b[n]: n in [1..m]]; // Vincenzo Librandi, Feb 01 2020

A000248 := proc(n) local k; add(k^(n-k)*binomial(n, k), k=0..n); end; # Robert FERREOL, Oct 11 2007
a:= proc(n) option remember; if n=0 then 1 else add(binomial(n-1, j) *(j+1) *a(n-1-j), j=0..n-1) fi end: seq(a(n), n=0..20); # Zerinvary Lajos, Mar 28 2009

CoefficientList[Series[Exp[x Exp[x]],{x,0,20}],x]*Table[n!,{n,0,20}]
a[0] = 1; a[1] = 1; a[n_] := a[n] = a[n - 1] + Sum[(Binomial[n - 1, j] + (n - 1) Binomial[n - 2, j]) a[j], {j, 0, n - 2}]; Table[a[n], {n, 0, 20}] (* David Callan, Oct 04 2013 *)
Flatten[{1,Table[Sum[Binomial[n,k]*(n-k)^k,{k,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, Jul 13 2014 *)
Table[Sum[BellY[n, k, Range[n]], {k, 0, n}], {n, 0, 20}] (* Vladimir Reshetnikov, Nov 09 2016 *)

a(n)=sum(k=0,n,binomial(n,k)*(n-k)^k); \\ Paul D. Hanna, Jun 26 2009

x='x+O('x^66); Vec(serlaplace(exp(x*exp(x)))) \\ Joerg Arndt, Oct 06 2013

# uses[bell_matrix from A264428]
B = bell_matrix(lambda k: k+1, 20)
print([sum(B.row(n)) for n in range(20)]) # Peter Luschny, Sep 03 2019

G.f.: Sum_{k>=0} x^k/(1-k*x)^(k+1). - Vladeta Jovovic, Oct 25 2003
a(n) = Sum_{k=0..n} C(n,k)*(n-k)^k. - Paul D. Hanna, Jun 26 2009
G.f.: G(0) where G(k) = 1 - x*(-1+2*k*x)^(2*k+1)/((x-1+2*k*x)^(2*k+2) - x*(x-1+2*k*x)^(4*k+4)/(x*(x-1+2*k*x)^(2*k+2) - (2*x-1+2*k*x)^(2*k+3)/G(k+1))) (continued fraction). - Sergei N. Gladkovskii, Jan 26 2013
E.g.f.: 1 + x/(1+x)*(G(0) - 1) where G(k) = 1 + exp(x)/(k+1)/(1-x/(x+(1)/G(k+1))) (continued fraction). - Sergei N. Gladkovskii, Feb 04 2013
Recurrence: a(0)=1, a(n) = Sum_{k=1..n} binomial(n-1,k-1)*k*a(n-k). - James East, Mar 30 2014
Asymptotics (Harris and Schoenfeld, 1968): a(n) ~ sqrt((r+1)/(2*Pi*(n+1)*(r^2+3*r+1))) * n! * exp((n+1)/(r+1)) / r^n, where r is the root of the equation r*(r+1)*exp(r) = n+1. - Vaclav Kotesovec, Jul 13 2014
a(n) = Sum_{k=0..n} A005727(k)*Stirling2(n, k). - Mélika Tebni, Jun 12 2022
More precise asymptotics: a(n) ~ n^(n + 1/2) / (sqrt(1 + 3*r + r^2) * exp(n - r*exp(r) + r/2) * r^(n + 1/2)), where r = 2*w - 1/(2*w) + 5/(8*w^2) - 19/(24*w^3) + 209/(192*w^4) - 763/(480*w^5) + 4657/(1920*w^6) - 6855/(1792*w^7) + 199613/(32256*w^8) + ... and w = LambertW(sqrt(n)/2). - Vaclav Kotesovec, Feb 20 2023

In view of the multiple appearances of this sequence, I replaced the definition with the simple exponential generating function. - N. J. A. Sloane, Apr 16 2018

A215703 A(n,k) is the n-th derivative of f_k at x=1, and f_k is the k-th of all functions that are representable as x^x^...^x with m>=1 x's and parentheses inserted in all possible ways; square array A(n,k), n>=0, k>=1, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 1, 1, 2, 0, 1, 1, 4, 3, 0, 1, 1, 2, 12, 8, 0, 1, 1, 6, 9, 52, 10, 0, 1, 1, 4, 27, 32, 240, 54, 0, 1, 1, 2, 18, 156, 180, 1188, -42, 0, 1, 1, 2, 15, 100, 1110, 954, 6804, 944, 0, 1, 1, 8, 9, 80, 650, 8322, 6524, 38960, -5112, 0, 1, 1, 6, 48, 56, 590, 4908, 70098, 45016, 253296, 47160, 0

Offset: 0

Alois P. Heinz, Aug 21 2012

A000081(m) distinct functions are representable as x^x^...^x with m>=1 x's and parentheses inserted in all possible ways.  Some functions are representable in more than one way, the number of valid parenthesizations is A000108(m-1).  The f_k are ordered, such that the number m of x's in f_k is a nondecreasing function of k.  The exact ordering is defined by the algorithm below.
The list of functions f_1, f_2, ... begins:
  | f_k : m : function (tree)  : representation(s)        : sequence |
  +-----+---+------------------+--------------------------+----------+
  | f_1 | 1 | x -> x           | x                        | A019590  |
  | f_2 | 2 | x -> x^x         | x^x                      | A005727  |
  | f_3 | 3 | x -> x^(x*x)     | (x^x)^x                  | A215524  |
  | f_4 | 3 | x -> x^(x^x)     | x^(x^x)                  | A179230  |
  | f_5 | 4 | x -> x^(x*x*x)   | ((x^x)^x)^x              | A215704  |
  | f_6 | 4 | x -> x^(x^x*x)   | (x^x)^(x^x), (x^(x^x))^x | A215522  |
  | f_7 | 4 | x -> x^(x^(x*x)) | x^((x^x)^x)              | A215705  |
  | f_8 | 4 | x -> x^(x^(x^x)) | x^(x^(x^x))              | A179405  |

			Square array A(n,k) begins:
  1,   1,    1,    1,     1,     1,     1,     1, ...
  1,   1,    1,    1,     1,     1,     1,     1, ...
  0,   2,    4,    2,     6,     4,     2,     2, ...
  0,   3,   12,    9,    27,    18,    15,     9, ...
  0,   8,   52,   32,   156,   100,    80,    56, ...
  0,  10,  240,  180,  1110,   650,   590,   360, ...
  0,  54, 1188,  954,  8322,  4908,  5034,  2934, ...
  0, -42, 6804, 6524, 70098, 41090, 47110, 26054, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns k=1-17, 37 give: A019590, A005727, A215524, A179230, A215704, A215522, A215705, A179405, A215706, A215707, A215708, A215709, A215691, A215710, A215643, A215629, A179505, A211205.
Rows n=0+1, 2-10 give: A000012, A215841, A215842, A215834, A215835, A215836, A215837, A215838, A215839, A215840.
Number of distinct values taken for m x's by derivatives n=1-10: A000012, A028310, A199085, A199205, A199296, A199883, A215796, A215971, A216062, A216403.
Main diagonal gives A306739.
Cf. A000081, A000108, A033917, A211192, A214569, A214570, A214571, A216041, A216281, A216349, A216350, A216351, A216368, A222379, A222380, A277537, A306679, A306710, A306726.

T:= proc(n) T(n):=`if`(n=1, [x], map(h-> x^h, g(n-1$2))) end:
g:= proc(n, i) option remember; `if`(i=1, [x^n], [seq(seq(
      seq(mul(T(i)[w[t]-t+1], t=1..j)*v, v=g(n-i*j, i-1)), w=
      combinat[choose]([$1..nops(T(i))+j-1], j)), j=0..n/i)])
    end:
f:= proc() local i, l; i, l:= 0, []; proc(n) while n>
      nops(l) do i:= i+1; l:= [l[], T(i)[]] od; l[n] end
    end():
A:= (n, k)-> n!*coeff(series(subs(x=x+1, f(k)), x, n+1), x, n):
seq(seq(A(n, 1+d-n), n=0..d), d=0..12);

T[n_] := If[n == 1, {x}, Map[x^#&, g[n - 1, n - 1]]];
g[n_, i_] := g[n, i] = If[i == 1, {x^n}, Flatten @ Table[ Table[ Table[ Product[T[i][[w[[t]] - t + 1]], {t, 1, j}]*v, {v, g[n - i*j, i - 1]}], {w, Subsets[ Range[ Length[T[i]] + j - 1], {j}]}], {j, 0, n/i}]];
f[n_] := Module[{i = 0, l = {}}, While[n > Length[l], i++; l = Join[l, T[i]]]; l[[n]]];
A[n_, k_] := n! * SeriesCoefficient[f[k] /. x -> x+1, {x, 0, n}];
Table[Table[A[n, 1+d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Nov 08 2019, after Alois P. Heinz *)

A179230 n-th derivative of x^(x^x) at x=1.

Original entry on oeis.org

1, 1, 2, 9, 32, 180, 954, 6524, 45016, 360144, 3023640, 27617832, 271481880, 2775329232, 31188079272, 350827041000, 4441125248640, 54110311240512, 765546040603584, 9938498593229568, 156934910753107200, 2128783325724881280, 37775147271084647424

Offset: 0

Henryk Trappmann (bo198214(AT)gmail.com), Jul 03 2010

sign

First term < 0: a(33) = -868875490363254484795699722301440.

Alois P. Heinz, Table of n, a(n) for n = 0..400

Cf. A005727. Column k=4 of A215703. Column k=3 of A277537.

Cf. A295103.

a:= n-> n!*coeff(series(subs(x=x+1, x^(x^x) ), x, n+1), x, n):
seq(a(n), n=0..30);  # Alois P. Heinz, Aug 20 2012

Table[ D[ x^(x^x), {x, n}] /. x -> 1, {n, 0, 20}] (* Robert G. Wilson v, Jul 12 2010 *)
NestList[ Factor[ D[ #1, x]] &, x^x^x, 20] /. x -> 1 (* Robert G. Wilson v, Aug 10 2010 *)
Range[0, 22]! CoefficientList[ Series[(1 + x)^(1 + x)^(1 + x), {x, 0, 22}], x] (* Robert G. Wilson v, Feb 03 2013 *)

E.g.f.: (x+1)^((x+1)^(x+1)). - Alois P. Heinz, Aug 21 2012

a(21)-a(22) from Alois P. Heinz, Aug 20 2012

A179405 n-th derivative of x^(x^(x^x)) at x=1.

Original entry on oeis.org

1, 1, 2, 9, 56, 360, 2934, 26054, 269128, 3010680, 37616880, 504880992, 7387701672, 115228447152, 1929016301016, 34194883090440, 643667407174464, 12757366498618176, 266426229010029696, 5830527979298793024, 133665090871032478080, 3197905600674249843840

Offset: 0

Robert G. Wilson v, Jul 13 2010

sign

First term < 0: a(329). - Alois P. Heinz, Sep 22 2015

Alois P. Heinz, Table of n, a(n) for n = 0..400

Cf. A005727, A179230, A295104.
Column k=8 of A215703.
Column k=4 of A277537.

a:= n-> n!*coeff(series(subs(x=x+1, x^(x^(x^x)) ), x, n+1), x, n):
seq(a(n), n=0..30);  # Alois P. Heinz, Aug 20 2012

f[n_] := D[ x^(x^(x^x)), {x, n}] /. x -> 1; Array[f, 18, 0]
Range[0, 21]! CoefficientList[ Series[(1 + x)^(1 + x)^(1 + x)^(1 + x), {x, 0, 21}], x] (* Robert G. Wilson v, Feb 03 2013 *)

E.g.f.: (x+1)^((x+1)^((x+1)^(x+1))). - Alois P. Heinz, Aug 23 2012

a(18)-a(21) from Alois P. Heinz, Aug 20 2012

A087761 Expansion of (1-x)^(1/(x-1)).

Original entry on oeis.org

1, 1, 4, 21, 140, 1130, 10674, 115206, 1396016, 18739080, 275712840, 4408612560, 76070179272, 1408041937848, 27816773482848, 583970117197320, 12978149959718400, 304310928180279360, 7506092106055537344

Offset: 0

Vladeta Jovovic, Oct 02 2003

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..439

Cf. A000248, A005727.

Table[Sum[BellY[n, k, Table[m! HarmonicNumber[m], {m, n}]], {k, 0, n}], {n, 0, 20}] (* Vladimir Reshetnikov, Nov 09 2016 *)

a(n)=if(n==0,1,(n-1)!*sum(k=0,n-1,(n-k)*sum(j=1,n-k,1/j)*a(k)/k!)) \\ Paul D. Hanna, Mar 17 2010; corrected Mar 19 2010

a(n) = Sum_{k=0..n} |Stirling1(n, k)|*A000248(k).
From Paul D. Hanna, Mar 17 2010: (Start)
E.g.f.: exp( Sum_{n>=1} H(n)*x^n ) where H(n) is the n-th harmonic number;
a(n) = (n-1)!*Sum_{k=0..n-1} (n-k)*H(n-k)*a(k)/k! for n>0 with a(0)=1. (End)
Empirical: a(n) = Sum_{i=0..n} binomial(n,i)*A005727(i)*(n-1)!/(i-1)! for n>0. - John M. Campbell, Dec 13 2016

A215524 n-th derivative of (x^x)^x at x=1.

Original entry on oeis.org

1, 1, 4, 12, 52, 240, 1188, 6804, 38960, 253296, 1654560, 11816640, 85816608, 668005728, 5240582592, 44667645120, 365989405440, 3494595006720, 28075694694912, 325862541872640, 2101211758356480, 39605981661066240, 48568198208302080, 7549838510211486720, -66667098077331572736

Offset: 0

Alois P. Heinz, Aug 14 2012

sign

Alois P. Heinz, Table of n, a(n) for n = 0..400

Column k=3 of A215703.

Cf. A005727, A179230, A179405, A215522.

a:= n-> n!*coeff(series(subs(x=x+1, (x^x)^x ), x, n+1), x, n):
seq(a(n), n=0..30);

m = 24; CoefficientList[((x+1)^(x+1))^(x+1) + O[x]^(m+1), x]*Range[0, m]! (* Jean-François Alcover, Feb 07 2021 *)

E.g.f.: ((x+1)^(x+1))^(x+1).

A277537 A(n,k) is the n-th derivative of the k-th tetration of x (power tower of order k) x^^k at x=1; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 2, 0, 0, 1, 1, 2, 3, 0, 0, 1, 1, 2, 9, 8, 0, 0, 1, 1, 2, 9, 32, 10, 0, 0, 1, 1, 2, 9, 56, 180, 54, 0, 0, 1, 1, 2, 9, 56, 360, 954, -42, 0, 0, 1, 1, 2, 9, 56, 480, 2934, 6524, 944, 0, 0, 1, 1, 2, 9, 56, 480, 4374, 26054, 45016, -5112, 0, 0

Offset: 0

Alois P. Heinz, Oct 19 2016

			Square array A(n,k) begins:
  1, 1,   1,    1,     1,     1,     1,     1, ...
  0, 1,   1,    1,     1,     1,     1,     1, ...
  0, 0,   2,    2,     2,     2,     2,     2, ...
  0, 0,   3,    9,     9,     9,     9,     9, ...
  0, 0,   8,   32,    56,    56,    56,    56, ...
  0, 0,  10,  180,   360,   480,   480,   480, ...
  0, 0,  54,  954,  2934,  4374,  5094,  5094, ...
  0, 0, -42, 6524, 26054, 47894, 60494, 65534, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
Eric Weisstein's World of Mathematics, Power Tower
Wikipedia, Knuth's up-arrow notation
Wikipedia, Tetration

Columns k=0..10 give A000007, A019590(n+1), A005727, A179230, A179405, A179505, A211205, A277538, A277539, A277540, A277541.
Rows n=0..1 give A000012, A057427.
Main diagonal gives A033917.
Cf. A215703, A277536, A295028.

f:= proc(n) f(n):= `if`(n=0, 1, (x+1)^f(n-1)) end:
A:= (n, k)-> n!*coeff(series(f(k), x, n+1), x, n):
seq(seq(A(n, d-n), n=0..d), d=0..14);
# second Maple program:
b:= proc(n, k) option remember; `if`(n=0, 1, `if`(k=0, 0,
      -add(binomial(n-1, j)*b(j, k)*add(binomial(n-j, i)*
      (-1)^i*b(n-j-i, k-1)*(i-1)!, i=1..n-j), j=0..n-1)))
    end:
A:= (n, k)-> b(n, min(k, n)):
seq(seq(A(n, d-n), n=0..d), d=0..14);

b[n_, k_] := b[n, k] = If[n==0, 1, If[k==0, 0, -Sum[Binomial[n-1, j]*b[j, k]*Sum[Binomial[n-j, i]*(-1)^i*b[n-j-i, k-1]*(i-1)!, {i, 1, n-j}], {j, 0, n-1}]]]; A[n_, k_] := b[n, Min[k, n]]; Table[A[n, d-n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, Jan 14 2017, adapted from 2nd Maple prog. *)

A(n,k) = [(d/dx)^n x^^k]_{x=1}.
E.g.f. of column k: (x+1)^^k.
A(n,k) = Sum_{i=0..min(n,k)} A277536(n,i).
A(n,k) = n * A295028(n,k) for n,k > 0.

A008296 Triangle of Lehmer-Comtet numbers of the first kind.

Original entry on oeis.org

1, 1, 1, -1, 3, 1, 2, -1, 6, 1, -6, 0, 5, 10, 1, 24, 4, -15, 25, 15, 1, -120, -28, 49, -35, 70, 21, 1, 720, 188, -196, 49, 0, 154, 28, 1, -5040, -1368, 944, 0, -231, 252, 294, 36, 1, 40320, 11016, -5340, -820, 1365, -987, 1050, 510, 45, 1, -362880, -98208, 34716, 9020, -7645, 3003, -1617, 2970, 825, 55, 1, 3628800

Offset: 1

N. J. A. Sloane

Triangle arising in the expansion of ((1+x)*log(1+x))^n.

Also the Bell transform of (-1)^(n-1)*(n-1)! if n>1 else 1 adding 1,0,0,0,... as column 0. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 16 2016

			Triangle begins:
   1;
   1,  1;
  -1,  3,   1;
   2, -1,   6,  1;
  -6,  0,   5, 10,  1;
  24,  4, -15, 25, 15, 1;
  ...

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 139.

Alois P. Heinz, Rows n = 1..141, flattened
H. W. Gould, A Set of Polynomials Associated with the Higher Derivatives of y = x^x, Rocky Mountain J. Math. Volume 26, Number 2 (1996), 615-625.
Tian-Xiao He and Yuanziyi Zhang, Centralizers of the Riordan Group, arXiv:2105.07262 [math.CO], 2021.
D. H. Lehmer, Numbers Associated with Stirling Numbers and x^x, Rocky Mountain J. Math., 15(2) 1985, pp. 461-475.

Cf. A039621 (second kind), A354795 (variant), A185164, A005727 (row sums), A298511 (central).
Columns: A045406 (column 2), A347276 (column 3), A345651 (column 4).
Diagonals: A000142, A000217, A059302.
Cf. A176118.

for n from 1 to 20 do for k from 1 to n do
printf(`%d,`, add(binomial(l,k)*k^(l-k)*Stirling1(n,l), l=k..n)) od: od:
# second program:
A008296 := proc(n, k) option remember; if k=1 and n>1 then (-1)^n*(n-2)! elif n=k then 1 else (n-1)*procname(n-2, k-1) + (k-n+1)*procname(n-1, k) + procname(n-1, k-1) end if end proc:
seq(print(seq(A008296(n, k), k=1..n)), n=1..7); # Mélika Tebni, Aug 22 2021

a[1, 1] = a[2, 1] = 1; a[n_, 1] = (-1)^n (n-2)!;
a[n_, n_] = 1; a[n_, k_] := a[n, k] = (n-1) a[n-2, k-1] + a[n-1, k-1] + (k-n+1) a[n-1,k]; Flatten[Table[a[n, k], {n, 1, 12}, {k, 1, n}]][[1 ;; 67]]
(* Jean-François Alcover, Apr 29 2011 *)

{T(n, k) = if( k<1 || k>n, 0, n! * polcoeff(((1 + x) * log(1 + x + x * O(x^n)))^k / k!, n))}; /* Michael Somos, Nov 15 2002 */

# uses[bell_matrix from A264428]
# Adds 1, 0, 0, 0, ... as column 0 at the left side of the triangle.
bell_matrix(lambda n: (-1)^(n-1)*factorial(n-1) if n>1 else 1, 7) # Peter Luschny, Jan 16 2016

E.g.f. for a(n, k): (1/k!)[ (1+x)*log(1+x) ]^k. - Len Smiley
Left edge is (-1)*n!, for n >= 2. Right edge is all 1's.
a(n+1, k) = n*a(n-1, k-1) + a(n, k-1) + (k-n)*a(n, k).
a(n, k) = Sum_{m} binomial(m, k)*k^(m-k)*Stirling1(n, m).
From Peter Bala, Mar 14 2012: (Start)
E.g.f.: exp(t*(1 + x)*log(1 + x)) = Sum_{n>=0} R(n,t)*x^n/n! = 1 + t*x + (t+t^2)x^2/2! + (-t+3*t^2+t^3)x^3/3! + .... Cf. A185164. The row polynomials R(n,t) are of binomial type and satisfy the recurrence R(n+1,t) = (t-n)*R(n,t) + t*d/dt(R(n,t)) + n*t*R(n-1,t) with R(0,t) = 1 and R(1,t) = t. Inverse array is A039621.
(End)
Sum_{k=0..n} (-1)^k * a(n,k) = A176118(n). - Alois P. Heinz, Aug 25 2021

More terms from James Sellers, Jan 26 2001

Edited by N. J. A. Sloane at the suggestion of Andrew Robbins, Dec 11 2007

A025566 a(n) = number of (s(0), s(1), ..., s(n)) such that every s(i) is a nonnegative integer, s(0) = 0, s(1) = 1, |s(i) - s(i-1)| <= 1 for i >= 2. Also a(n) = sum of numbers in row n+1 of the array T defined in A026105. Also a(n) = T(n,n), where T is the array defined in A025564.

Original entry on oeis.org

1, 1, 1, 3, 8, 22, 61, 171, 483, 1373, 3923, 11257, 32418, 93644, 271219, 787333, 2290200, 6673662, 19478091, 56930961, 166613280, 488176938, 1431878079, 4203938697, 12353600427, 36331804089, 106932444885, 314946659951, 928213563878

Offset: 0

Clark Kimberling

nonn

a(n+1) is the number of Motzkin (2n)-paths whose last weak valley occurs immediately after step n. A weak valley in a Motzkin path (A001006) is an interior vertex whose following step has nonnegative slope and whose preceding step has nonpositive slope. For example, the weak valleys in the Motzkin path F.UF.FD.UD occur after the first, third and fifth steps as indicated by the dots (U=upstep of slope 1, D=downstep of slope -1, F=flatstep of slope 0) and, with n=2, a(3)=3 counts FFUD, UDUD, UFFD. - David Callan, Jun 07 2006

Starting with offset 2: (1, 3, 8, 22, 61, 171, 483, ...), = row sums of triangle A136537. - Gary W. Adamson, Jan 04 2008

Michael De Vlieger, Table of n, a(n) for n = 0..2100
Jean-Luc Baril, Richard Genestier, Sergey Kirgizov, Pattern distributions in Dyck paths with a first return decomposition constrained by height, arXiv:1911.03119 [math.CO], 2019.
C. Dalfó, M. A. Fiol, and N. López, New results for the Mondrian art problem, arXiv:2007.09639 [math.CO], 2020.
D. E. Davenport, L. W. Shapiro and L. C. Woodson, The Double Riordan Group, The Electronic Journal of Combinatorics, 18(2) (2012), #P33. - From _N. J. A. Sloane_, May 11 2012
Christian Krattenthaler, Daniel Yaqubi, Some determinants of path generating functions, II, arXiv:1802.05990 [math.CO], 2018; Adv. Appl. Math. 101 (2018), 232-265.
Donatella Merlini, Massimo Nocentini, Algebraic Generating Functions for Languages Avoiding Riordan Patterns, Journal of Integer Sequences, Vol. 21 (2018), Article 18.1.3.

First differences of A026135. Row sums of triangle A026105.
Pairwise sums of A005727. Column k=2 in A115990.
Cf. A001791, A132816.
Cf. A136537.

List([0..30],i->Sum([0..Int(i/2)],k->Binomial(i-2,k)*Binomial(i-k,k))); # Muniru A Asiru, Mar 09 2019

seq( sum('binomial(i-2,k)*binomial(i-k,k)', 'k'=0..floor(i/2)), i=0..30 ); # Detlef Pauly (dettodet(AT)yahoo.de), Nov 09 2001

CoefficientList[Series[x+(2x(x-1))/(1-3x-Sqrt[1-2x-3x^2]),{x,0,30}],x] (* Harvey P. Dale, Jun 12 2016 *)

G.f.: x + 2*x*(x-1)/(1-3x-sqrt(1-2x-3x^2)); for n > 1, first differences of the "directed animals" sequence A005773: a(n) = A005773(n) - A005773(n-1). - Emeric Deutsch, Aug 16 2002
Starting (1, 3, 8, 22, 61, 171, ...) gives the inverse binomial transform of A001791 starting (1, 4, 15, 56, 210, 792, ...). - Gary W. Adamson, Sep 01 2007
a(n) is the sum of the (n-2)-th row of triangle A131816. - Gary W. Adamson, Sep 01 2007
D-finite with recurrence n*a(n) +(-3*n+2)*a(n-1) +(-n+2)*a(n-2) +3*(n-4)*a(n-3)=0. - R. J. Mathar, Sep 15 2020

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