A000169 -id:A000169 - cp's OEIS frontend

A055302 Triangle of number of labeled rooted trees with n nodes and k leaves, n >= 1, 1 <= k <= n.

1, 2, 0, 6, 3, 0, 24, 36, 4, 0, 120, 360, 140, 5, 0, 720, 3600, 3000, 450, 6, 0, 5040, 37800, 54600, 18900, 1302, 7, 0, 40320, 423360, 940800, 588000, 101136, 3528, 8, 0, 362880, 5080320, 16087680, 15876000, 5143824, 486864, 9144, 9, 0, 3628800

Offset: 1

Christian G. Bower, May 11 2000

Beginning with the second row, dividing each row by n gives the mirror of row n-1 of A141618. Under the exponential transform, the mirror of A141618 is generated, relating the number of connected graphs here to the number of disconnected graphs associated with A141618 (cf. A127671 and A036040). - Tom Copeland, Oct 25 2014

			Triangle begins
     1,
     2,     0;
     6,     3,     0;
    24,    36,     4,     0;
   120,   360,   140,     5,    0;
   720,  3600,  3000,   450,    6, 0;
  5040, 37800, 54600, 18900, 1302, 7, 0;

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 313.

Alois P. Heinz, Rows n = 1..141, flattened
N. J. A. Sloane, Transforms
Index entries for sequences related to rooted trees

Row sums give A000169. Columns 1 through 12: A000142, A055303-A055313. Cf. A055314.

Cf. A248120 for a natural refinement.

T:= (n, k)-> (n!/k!)*Stirling2(n-1, n-k):
seq(seq(T(n, k), k=1..n), n=1..10);  # Alois P. Heinz, Nov 13 2013

Table[Table[n!/k! StirlingS2[n-1,n-k], {k,1,n}], {n,0,10}]//Grid  (* Geoffrey Critzer, Dec 01 2012 *)

A055302(n,k)=n!/k!*stirling(n-1, n-k,2);
for(n=1,10,for(k=1,n,print1(A055302(n,k),", "));print());
\\ Joerg Arndt, Oct 27 2014

E.g.f. (relative to x) satisfies: A(x,y) = xy + x*exp(A(x,y)) - x. Divides by n and shifts up under exponential transform.
T(n,k) = (n!/k!)*Stirling2(n-1, n-k). - Vladeta Jovovic, Jan 28 2004
T(n,k) = A055314(n,k)*(n-k) + A055314(n,k+1)*(k+1). The first term is the number of such trees with root degree > 1 while the second term is the number of such trees with root degree = 1. This simplifies to the above formula by Vladeta Jovovic. - Geoffrey Critzer, Dec 01 2012
E.g.f.: G(x,t) = log[1 + t * N(x*t,1/t)], where N(x,t) is the e.g.f. of A141618. Also, G(x*t,1/t)= log[1 + N(x,t)/t] is the comp. inverse in x of x / [1 + t * (e^x - 1)]. - Tom Copeland, Oct 26 2014

A063170 Schenker sums with n-th term.

Original entry on oeis.org

1, 2, 10, 78, 824, 10970, 176112, 3309110, 71219584, 1727242866, 46602156800, 1384438376222, 44902138752000, 1578690429731402, 59805147699103744, 2428475127395631750, 105224992014096760832, 4845866591896268695010, 236356356027029797011456

Offset: 0

Marijke van Gans (marijke(AT)maxwellian.demon.co.uk)

Urn, n balls, with replacement: how many selections if we stop after a ball is chosen that was chosen already? Expected value is a(n)/n^n.
Conjectures: The exponent in the power of 2 in the prime factorization of a(n) (its 2-adic valuation) equals 1 if n is odd and equals n - A000120(n) if n is even. - Gerald McGarvey, Nov 17 2007, Jun 29 2012
Amdeberhan, Callan, and Moll (2012) have proved McGarvey's conjectures. - Jonathan Sondow, Jul 16 2012
a(n), for n >= 1, is the number of colored labeled mappings from n points to themselves, where each component is one of two colors. - Steven Finch, Nov 28 2021

			a(4) = (1*2*3*4) + 4*(2*3*4) + 4*4*(3*4) + 4*4*4*(4) + 4*4*4*4.
G.f. = 1 + 2*x + 10*x^2 + 78*x^3 + 824*x^4 + 10970*x^5 + 176112*x^6 + ...

D. E. Knuth, The Art of Computer Programming, 3rd ed. 1997, Vol. 1, Addison-Wesley, p. 123, Exercise Section 1.2.11.3 18.

G. C. Greubel, Table of n, a(n) for n = 0..385
Max Alekseyev, Recursion for A063170, answer to question on MathOverflow (2025).
T. Amdeberhan, D. Callan, and V. Moll, p-adic analysis and combinatorics of truncated exponential sums, preprint, 2012.
T. Amdeberhan, D. Callan and V. Moll, Valuations and combinatorics of truncated exponential sums, INTEGERS 13 (2013), #A21.
Steven Finch, Rounds, Color, Parity, Squares, arXiv:2111.14487 [math.CO], 2021.
Helmut Prodinger, An identity conjectured by Lacasse via the tree function, Electronic Journal of Combinatorics, 20(3) (2013), #P7.
David M. Smith and Geoffrey Smith, Tight Bounds on Information Leakage from Repeated Independent Runs, 2017 IEEE 30th Computer Security Foundations Symposium (CSF).
Marijke van Gans, Schenker sums
Eric Weisstein, Exponential Sum Function.

Cf. A000312, A134095, A090878, A036505, A120266, A214402, A219546 (Schenker primes).

seq(simplify(GAMMA(n+1,n)*exp(n)),n=0..20); # Vladeta Jovovic, Jul 21 2005

a[n_] := Round[ Gamma[n+1, n]*Exp[n]]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Feb 16 2012, after Vladeta Jovovic *)
a[ n_] := If[ n < 1, Boole[n == 0], n! Sum[ n^k / k!, {k, 0, n}]]; (* Michael Somos, Jun 05 2014 *)
a[ n_] := If[ n < 0, 0, n! Normal[ Exp[x] + x O[x]^n] /. x -> n]; (* Michael Somos, Jun 05 2014 *)

{a(n) = if( n<0, 0, n! * sum( k=0, n, n^k / k!))};

{a(n) = sum( k=0, n, binomial(n, k) * k^k * (n - k)^(n - k))}; /* Michael Somos, Jun 09 2004 */

for(n=0,17,print1(round(intnum(x=0,[oo,1],exp(-x)*(n+x)^n)),", ")) \\ Gerald McGarvey, Nov 17 2007

from math import comb
def A063170(n): return (sum(comb(n,k)*(n-k)**(n-k)*k**k for k in range(1,(n+1>>1)))<<1) + (0 if n&1 else comb(n,m:=n>>1)*m**n) + (n**n<<1) if n else 1 # Chai Wah Wu, Apr 26 2023

10 for N=1 to 42: T=N^N: S=T
20 for K=N to 1 step -1: T/=N: T*=K: S+=T: next K
30 print N,S: next N

a(n) = Sum_{k=0..n} n^k n!/k!.
a(n)/n! = Sum_{k=0..n} n^k/k!. (First n+1 terms of e^n power series.)
a(n) = A063169(n) + n^n.
E.g.f.: 1/(1-T)^2, where T=T(x) is Euler's tree function (see A000169).
E.g.f.: 1 / (1 - F), where F = F(x) is the e.g.f. of A003308. - Michael Somos, May 27 2012
a(n) = Sum_{k=0..n} binomial(n,k)*(n+k)^k*(-k)^(n-k). - Vladeta Jovovic, Oct 11 2007
Asymptotics of the coefficients: sqrt(Pi*n/2)*n^n. - N-E. Fahssi, Jan 25 2008
a(n) = A120266(n)*A214402(n) for n > 0. - Jonathan Sondow, Jul 16 2012
a(n) = Integral_{0..oo} exp(-x) * (n + x)^n dx. - Michael Somos, May 18 2004
a(n) = Integral_{0..oo} exp(-x)*(1+x/n)^n dx * n^n = A090878(n)/A036505(n-1) * n^n. - Gerald McGarvey, Nov 17 2007
EXP-CONV transform of A000312. - Tilman Neumann, Dec 13 2008
a(n) = n! * [x^n] exp(n*x)/(1 - x). - Ilya Gutkovskiy, Sep 23 2017
a(n) = (n+1)! - Sum_{k=0..n-1} binomial(n, k)*a(k)*(-k)^(n-k) for n > 0 with a(0) = 1 (see Max Alekseyev link). - Mikhail Kurkov, Jan 14 2025

A003992 Square array read by upwards antidiagonals: T(n,k) = n^k for n >= 0, k >= 0.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 4, 1, 0, 1, 4, 9, 8, 1, 0, 1, 5, 16, 27, 16, 1, 0, 1, 6, 25, 64, 81, 32, 1, 0, 1, 7, 36, 125, 256, 243, 64, 1, 0, 1, 8, 49, 216, 625, 1024, 729, 128, 1, 0, 1, 9, 64, 343, 1296, 3125, 4096, 2187, 256, 1, 0, 1, 10, 81, 512, 2401, 7776, 15625, 16384, 6561, 512, 1, 0

Offset: 0

Marc LeBrun

If the array is transposed, T(n,k) is the number of oriented rows of n colors using up to k different colors. The formula would be T(n,k) = [n==0] + [n>0]*k^n. The generating function for column k would be 1/(1-k*x). For T(3,2)=8, the rows are AAA, AAB, ABA, ABB, BAA, BAB, BBA, and BBB. - Robert A. Russell, Nov 08 2018

T(n,k) is the number of multichains of length n from {} to [k] in the Boolean lattice B_k. - Geoffrey Critzer, Apr 03 2020

			Rows begin:
[1, 0,  0,   0,    0,     0,      0,      0, ...],
[1, 1,  1,   1,    1,     1,      1,      1, ...],
[1, 2,  4,   8,   16,    32,     64,    128, ...],
[1, 3,  9,  27,   81,   243,    729,   2187, ...],
[1, 4, 16,  64,  256,  1024,   4096,  16384, ...],
[1, 5, 25, 125,  625,  3125,  15625,  78125, ...],
[1, 6, 36, 216, 1296,  7776,  46656, 279936, ...],
[1, 7, 49, 343, 2401, 16807, 117649, 823543, ...], ...

T. D. Noe, Rows n = 0..50 of triangle, flattened

Rows 0-49 are A000007, A000012, A000079, A000244, A000302, A000351, A000400, A000420, A001018, A001019, A011557, A001020, A001021, A001022, A001023, A001024, A001025, A001026, A001027, A001029, A009964-A009992, A087752.
Columns 0-26 are A000012, A001477, A000290, A000578, A000583, A000584, A001014, A001015, A001016, A001017, A008454, A008455, A008456, A010801-A010813, A089081.
Main diagonal is A000312. Other diagonals include A000169, A007778, A000272, A008788. Antidiagonal sums are in A026898.
Cf. A099555.
Transpose is A004248. See A051128, A095884, A009999 for other versions.
Cf. A277504 (unoriented), A293500 (chiral).

[[(n-k)^k: k in [0..n]]: n in [0..10]]; // G. C. Greubel, Nov 08 2018

Table[If[k == 0, 1, (n - k)^k], {n, 0, 11}, {k, 0, n}]//Flatten

T(n,k) = (n-k)^k \\ Charles R Greathouse IV, Feb 07 2017

E.g.f.: Sum T(n,k)*x^n*y^k/k! = 1/(1-x*exp(y)). - Paul D. Hanna, Oct 22 2004

E.g.f.: Sum T(n,k)*x^n/n!*y^k/k! = e^(x*e^y). - Franklin T. Adams-Watters, Jun 23 2006

More terms from David W. Wilson

Edited by Paul D. Hanna, Oct 22 2004

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

A052182 Determinant of n X n matrix whose rows are cyclic permutations of 1..n.

Original entry on oeis.org

1, -3, 18, -160, 1875, -27216, 470596, -9437184, 215233605, -5500000000, 155624547606, -4829554409472, 163086595857367, -5952860799406080, 233543408203125000, -9799832789158199296, 437950726881001816329, -20766159817517617053696, 1041273502979112415328410

Offset: 1

Henry M. Gunn High School Mathematical Circle (Joshua Zucker), Jan 26 2000

Each row is a cyclic shift to the right by one place of the previous row. See the example below. - N. J. A. Sloane, Jan 07 2019
|a(n)| = number of labeled mappings from n points to themselves (endofunctions) with an odd number of cycles. - Vladeta Jovovic, Mar 30 2006
|a(n)| = number of functions from {1,2,...,n}->{1,2,...,n} such that of all recurrent elements the least is always mapped to the greatest. - Geoffrey Critzer, Aug 29 2013

			a(3) = 18 because this is the determinant of [(1,2,3), (3,1,2), (2,3,1) ].

T. D. Noe, Table of n, a(n) for n = 1..100
P. J. Cameron and P. Cara, Independent generating sets and geometries for symmetric groups, J. Algebra, Vol. 258, no. 2 (2002), 641-650.

Cf. A000312, A070896, A060281, A060435.

1,seq(LinearAlgebra:-Determinant(Matrix(n,shape=Circulant[$1..n])),n=2..30); # Robert Israel, Aug 31 2014

f[n_] := Det[ Table[ RotateLeft[ Range@ n, -j], {j, 0, n - 1}]]; Array[f, 19] (* or *)
f[n_] := (-1)^(n - 1)*n^(n - 2)*(n^2 + n)/2; Array[f, 19]
(* Robert G. Wilson v, Aug 31 2014 *)
Table[Det[Table[RotateRight[Range[k],n],{n,0,k-1}]],{k,30}] (* Harvey P. Dale, Jun 20 2024 *)

(1+n)^(n-1)*binomial(n+2,n)*(-1)^(n) $ n=0..16 // Zerinvary Lajos, Apr 01 2007

a(n) = (n+1)*(-n)^(n-1)/2; \\ Altug Alkan, Dec 17 2017

a(n) = (-1)^(n-1) * n^(n-2) * (n^2 + n)/2.
E.g.f.[A052182] = E.g.f.[A000312] * E.g.f.[A000272], so A052182(unsigned) is "tree-like". E.g.f.: (T-T^2/2)/(1-T), where T=T(x) is Euler's tree function (see A000169). E.g.f. for signed sequence: (W+W^2/2)/(1+W), where W=W(x)=-T(-x) is the Lambert W function. - Len Smiley, Dec 13 2001
Conjecture: a(n) = -Res( f(n), x^n - 1), where Res is the resultant and f(n) = Sum_{k=1..n} k*x^k. - Benedict W. J. Irwin, Dec 07 2016

More terms from James Sellers, Jan 31 2000

A052750 a(n) = (2*n + 1)^(n - 1).

Original entry on oeis.org

1, 1, 5, 49, 729, 14641, 371293, 11390625, 410338673, 16983563041, 794280046581, 41426511213649, 2384185791015625, 150094635296999121, 10260628712958602189, 756943935220796320321, 59938945498865420543457

Offset: 0

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

a(n+1) is the number of labeled incomplete ternary trees on n vertices in which each left child has a larger label than its parent. - Brian Drake, Jul 28 2008
Put a(0) = 1. For n > 0, let x(n,k) = 2*cos((2*k-1)*Pi/(2*n+1)), k=1..n. Define the recurrences S(n;0,x(n,k)) = 1, S(n;1,x(n,k)) = x(n,k), S(n;r,x(n,k)) = x(n,k)*S(n;r-1,x(n,k)) - S(n;r-2,x(n,k)), r > 1 an integer, k=1..n. CONJECTURE: For n > 0, a(n) = Product_{k=1..n} (Sum_{m=0..n-1} S(n;2*m,x(n,k))^2). - L. Edson Jeffery, Sep 11 2013
From Wolfdieter Lang, Dec 16 2013: (Start)
Discriminants of the first difference of Chebyshev S-polynomials.
The coefficient table for the first difference polynomials P(n, x) = S(n, x) - S(n-1, x), n >= 0, S(-1, x) = 0, with the Chebyshev S polynomials (see A049310), is given in A130777.
For the discriminant of a polynomial in terms of the square of a determinant of a Vandermonde matrix build from the zeros of the polynomial see, e.g., A127670.
For the proof that D(n) := discriminant(P(n,x)) = (2*n + 1)^(n - 1), n >= 1, use the formula given e.g., in the Rivlin reference, p. 218, Theorem 5.13, eq. (5.3), namely D(n) = (-1)^(n*(n-1)/2)*Product_{j=1..n} P'(n, x(n,j)), with the zeros x(n,j) = -2*cos(2*Pi*j/(2*n+1)) of P(n, x) (see A130777). P'(n, x(n,j)) = (2*n+1)*P(n-1, x(n,j))/(2*sin(Pi*j/(2*n+1))*2*cos(Pi*j/(2*n+1)))^2. P(n-1, x(n,j)) = (-1)^(n+j)*2*cos(Pi*j/(2*n+1)). Product_{j=1..n} 2*sin(Pi*j/(2*n+1)) = 2*n+1 (see the Oct 10 2013 formula in A005408. Product_{j=1..n} 2*cos(Pi*j/(2*n+1)) = 1, because S(2*n, 0) = (-1)^n.
(End)
a(n) is the number of labeled 2-trees with n+2 vertices, rooted at a given edge. - Nikos Apostolakis, Nov 30 2018
a(n) is also the number of 2-trees with n labeled triangles and with a distinguished oriented edge. - Nikos Apostolakis, Dec 14 2018

			Discriminant: n=4: P(4, x) = 1 + 2*x - 3*x^2 - x^3 + x^4 with the zeros x[1] = -2*cos((2/9)*Pi), x[2] = -2*cos((4/9)*Pi), x[3] = 1, x[4] = 2*cos((1/9)*Pi). D(4) = (Det(Vandermonde(4,[x[1],x[2],x[3],x[4]])))^2 = 729 = a(4). - _Wolfdieter Lang_, Dec 16 2013

L. W. Beineke, and J. W. Moon, Several proofs of the number of labelled 2-dimensional trees, In "Proof Techniques in Graph Theory" (F. Harary editor). Academic Press, New York, 1969, pp. 11-20.
Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990.

G. C. Greubel, Table of n, a(n) for n = 0..350
T. Fowler, I. Gessel, G. Labelle, and P. Leroux, The specification of 2-trees, Adv. Appl. Math. 28 (2) (2002) 145-168, eq. (9).
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 706
Henri Muehle and Philippe Nadeau, A Poset Structure on the Alternating Group Generated by 3-Cycles, arXiv:1803.00540 [math.CO], 2018.
J.-C. Novelli and J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014.
Index entries for sequences related to Chebyshev polynomials.

Cf. A127670, A130777, A000169, A052752, A052774, A052782, A000272, A001705.

List([0..20],n->(2*n+1)^(n-1)); # Muniru A Asiru, Dec 05 2018

[(2*n+1)^(n-1) : n in [0..20]]; // Wesley Ivan Hurt, Jan 20 2017

spec := [S,{B=Prod(Z,S,S),S=Set(B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

max = 16; (Series[Exp[-1/2*ProductLog[-2*x]], {x, 0, max}] // CoefficientList[#, x] & ) * Range[0, max]! (* Jean-François Alcover, Jun 20 2013 *)
Table[(2*n+1)^(n-1),{n,0,20}] (* Harvey P. Dale, Jul 14 2025 *)

a(n)=(2*n+1)^(n-1) \\ Charles R Greathouse IV, Nov 20 2011

{a(n)=local(A=1+x);for(i=1,21,A=sqrt(1+2*sum(n=1,21,x^(2*n-1)/(2*n-1)!*A^(4*n-1))+x*O(x^n)));n!*polcoeff(A,n)} \\ Paul D. Hanna, Sep 07 2012

for n in range(0, 20): print((2*n + 1)**(n - 1), end=', ') # Stefano Spezia, Dec 01 2018

E.g.f.: exp(-1/2*W(-2*x)), where W is Lambert's W function.
E.g.f. satisfies: A(x) = sqrt(1 + 2*Sum_{n>=1} x^(2*n-1)/(2*n-1)! * A(x)^(4*n-1)). - Paul D. Hanna, Sep 07 2012
E.g.f. satisfies: A(x) = 1/A(-x*A(x)^4). - Paul D. Hanna, Sep 07 2012
a(n) = discriminant of P(n,x) = S(n,x) - S(n-1,x), n >= 1, with the Chebyshev S polynomials from A049310. For the proof see the comment above. a(n) is also the discriminant of S(n,x) + S(n-1,x) = (-1)^n*(S(n,-x) - S(n-1,-x)). - Wolfdieter Lang, Dec 16 2013
From Peter Bala, Dec 19 2013: (Start)
The e.g.f. A(x) = 1 + x + 5*x^2/2! + 49*x^3/3! + 729*x^4/4! + ... satisfies:
1) A(x*exp(-2*x)) = exp(x) = 1/A(-x*exp(2*x));
2) A^2(x) = 1/x*series reversion(x*exp(-2*x));
3) A(x^2) = 1/x*series reversion(x*exp(-x^2));
4) A(x) = exp(x*A(x)^2). (End)
E.g.f.: sqrt(-LambertW(-2*x)/(2*x)). - Vaclav Kotesovec, Dec 07 2014
Related to A001705 by Sum_{n >= 1} a(n)*x^n/n! = series reversion( 1/(1 + x)^2*log(1 + x) ) = series reversion(x - 5*x^2/2! + 26*x^3/3! - 154*x^4/4! + ...). Cf. A000272, A052752, A052774, A052782. - Peter Bala, Jun 15 2016
From Peter Bala, Dec 13 2022: (Start)
The e.g.f. A(x) = 1/x * series reversion of x^2/T(x), where the tree function T(x) = Sum_{n >= 1} n^(n-1)*x^n/n!. See A000169.
For c in C, A(x)^c = 1 + Sum_{n >= 1} c*(2*n + c)^(n-1)*x^n/n!.
First derivative A'(x) = A(x)^3/(1 - 2*x*A(x)^2).
Series reversion of (1 - A(-z)) = -log(1 - z)/(1 - z)^2 is the e.g.f. of A001705.
1/z * series reversion of z/A(z) = 1 + z + 7*z^2/2! + (10^2)*z^3/3! + (13^3)*z^4/4! + ... is the e.g.f. of A052752.
1/z * series reversion of z/A(z^2) = 1 + z^2 + 9*z^4/2! + (13^2)*z^6/3! + (17^3)*z^8/4! + ... = Sum_{n >= 0} A052774(n)*z^(2*n)/n!.
1/z * series reversion of z/A(z^3) = 1 + z^3 + 11*z^6/2! + (16^2)*z^9/3! + (21^3)*z^12/4! + ... = Sum_{n >= 0} A052782(n)*z^(3*n)/n!.
1/z * series reversion of z/A(z)^2 = A(2*z) = 2*Sum_{n >= 0} (4*n + 2)^(n-1)*z^n/n!.
1/z * series reversion of z/A(z)^k = k*Sum_{n >= 0} ((k+2)*n + k)^(n-1)*z^n/n!. (End)
a(n) = Sum_{k=1..n} (-1)^(n-k)*(n+k)^(n-1)*binomial(n,k-1), a(0)=1. - Vladimir Kruchinin, Aug 14 2025

Better description from Vladeta Jovovic, Sep 02 2003

A060281 Triangle T(n,k) read by rows giving number of labeled mappings (or functional digraphs) from n points to themselves (endofunctions) with exactly k cycles, k=1..n.

Original entry on oeis.org

1, 3, 1, 17, 9, 1, 142, 95, 18, 1, 1569, 1220, 305, 30, 1, 21576, 18694, 5595, 745, 45, 1, 355081, 334369, 113974, 18515, 1540, 63, 1, 6805296, 6852460, 2581964, 484729, 49840, 2842, 84, 1, 148869153, 158479488, 64727522, 13591116, 1632099, 116172, 4830, 108, 1

Offset: 1

Vladeta Jovovic, Apr 09 2001

Also called sagittal graphs.
T(n,k)=1 iff n=k (counts the identity mapping of [n]). - Len Smiley, Apr 03 2006
Also the coefficients of the tree polynomials t_{n}(y) defined by (1-T(z))^(-y) = Sum_{n>=0} t_{n}(y) (z^n/n!) where T(z) is Cayley's tree function T(z) = Sum_{n>=1} n^(n-1) (z^n/n!) giving the number of labeled trees A000169. - Peter Luschny, Mar 03 2009

			Triangle T(n,k) begins:
        1;
        3,       1;
       17,       9,       1;
      142,      95,      18,      1;
     1569,    1220,     305,     30,     1;
    21576,   18694,    5595,    745,    45,    1;
   355081,  334369,  113974,  18515,  1540,   63,  1;
  6805296, 6852460, 2581964, 484729, 49840, 2842, 84, 1;
  ...
T(3,2)=9: (1,2,3)--> [(2,1,3),(3,2,1),(1,3,2),(1,1,3),(1,2,1), (1,2,2),(2,2,3),(3,2,3),(1,3,3)].
From _Peter Luschny_, Mar 03 2009: (Start)
  Tree polynomials (with offset 0):
  t_0(y) = 1;
  t_1(y) = y;
  t_2(y) = 3*y + y^2;
  t_3(y) = 17*y + 9*y^2 + y^3; (End)

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983.
W. Szpankowski. Average case analysis of algorithms on sequences. John Wiley & Sons, 2001. - Peter Luschny, Mar 03 2009

Alois P. Heinz, Rows n = 1..141, flattened
Julia Handl and Joshua Knowles, An Investigation of Representations and Operators for Evolutionary Data Clustering with a Variable Number of Clusters, in Parallel Problem Solving from Nature-PPSN IX, Lecture Notes in Computer Science, Volume 4193/2006, Springer-Verlag. [From _N. J. A. Sloane_, Jul 09 2009]
D. E. Knuth, Convolution polynomials, The Mathematica J., 2 (1992), 67-78.
D. E. Knuth and B. Pittel, A recurrence related to trees, Proceedings of the American Mathematical Society, 105(2):335-349, 1989. [From _Peter Luschny_, Mar 03 2009]
J. Riordan, Enumeration of Linear Graphs for Mappings of Finite Sets, Ann. Math. Stat., 33, No. 1, Mar. 1962, pp. 178-185.
David M. Smith and Geoffrey Smith, Tight Bounds on Information Leakage from Repeated Independent Runs, 2017 IEEE 30th Computer Security Foundations Symposium (CSF).

Columns k=1-10 give: A001865, A065456, A273434, A273435, A273436, A273437, A273438, A273439, A273440, A273441.
Row sums: A000312.
Main diagonal and first lower diagonal give: A000012, A045943.
Cf. A000169, A190314, A242027, A209324, A273442, A347999.

A060281:= func< n,k | (&+[Binomial(n-1,j)*n^(n-1-j)*(-1)^(k+j+1)*StirlingFirst(j+1,k): j in [0..n-1]]) >;
[A060281(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Nov 06 2024

with(combinat):T:=array(1..8,1..8):for m from 1 to 8 do for p from 1 to m do T[m,p]:=sum(binomial(m-1,k)*m^(m-1-k)*(-1)^(p+k+1)*stirling1(k+1,p),k=0..m-1); print(T[m,p]) od od; # Len Smiley, Apr 03 2006
From Peter Luschny, Mar 03 2009: (Start)
T := z -> sum(n^(n-1)*z^n/n!,n=1..16):
p := convert(simplify(series((1-T(z))^(-y),z,12)),'polynom'):
seq(print(coeff(p,z,i)*i!),i=0..8); (End)

t=Sum[n^(n-1) x^n/n!,{n,1,10}];
Transpose[Table[Rest[Range[0, 10]! CoefficientList[Series[Log[1/(1 - t)]^n/n!, {x, 0, 10}], x]], {n,1,10}]]//Grid (* Geoffrey Critzer, Mar 13 2011*)
Table[k! SeriesCoefficient[1/(1 + ProductLog[-t])^x, {t, 0, k}, {x, 0, j}], {k, 10}, {j, k}] (* Jan Mangaldan, Mar 02 2013 *)

@CachedFunction
def A060281(n,k): return sum(binomial(n-1,j)*n^(n-1-j)*stirling_number1(j+1,k) for j in range(n))
flatten([[A060281(n,k) for k in range(1,n+1)] for n in range(1,13)]) # G. C. Greubel, Nov 06 2024

E.g.f.: 1/(1 + LambertW(-x))^y.
T(n,k) = Sum_{j=0..n-1} C(n-1,j)*n^(n-1-j)*(-1)^(k+j+1)*A008275(j+1,k) = Sum_{j=0..n-1} binomial(n-1,j)*n^(n-1-j)*s(j+1,k). [Riordan] (Note: s(m,p) denotes signless Stirling cycle number (first kind), A008275 is the signed triangle.) - Len Smiley, Apr 03 2006
T(2*n, n) = A273442(n), n >= 1. -  Alois P. Heinz, May 22 2016
From Alois P. Heinz, Dec 17 2021: (Start)
Sum_{k=1..n} k * T(n,k) = A190314(n).
Sum_{k=1..n} (-1)^(k+1) * T(n,k) = A000169(n) for n>=1. (End)

A134991 Triangle of Ward numbers T(n,k) read by rows.

Original entry on oeis.org

1, 1, 3, 1, 10, 15, 1, 25, 105, 105, 1, 56, 490, 1260, 945, 1, 119, 1918, 9450, 17325, 10395, 1, 246, 6825, 56980, 190575, 270270, 135135, 1, 501, 22935, 302995, 1636635, 4099095, 4729725, 2027025, 1, 1012, 74316, 1487200, 12122110, 47507460, 94594500, 91891800, 34459425

Offset: 1

Tom Copeland, Feb 05 2008

This is the triangle of associated Stirling numbers of the second kind, A008299, read along the diagonals.
This is also a row-reversed version of A181996 (with an additional leading 1) - see the table on p. 92 in the Ward reference. A134685 is a refinement of the Ward table.
The first and second diagonals are A001147 and A000457 and appear in the diagonals of several OEIS entries. The polynomials also appear in Carlitz (p. 85), Drake et al. (p. 8) and Smiley (p. 7).
First few polynomials (with a different offset) are
  P(0,t) = 0
  P(1,t) = 1
  P(2,t) = t
  P(3,t) = t + 3*t^2
  P(4,t) = t + 10*t^2 + 15*t^3
  P(5,t) = t + 25*t^2 + 105*t^3 + 105*t^4
These are the "face" numbers of the tropical Grassmannian G(2,n),related to phylogenetic trees (with offset 0 beginning with P(2,t)). Corresponding h-vectors are A008517. - Tom Copeland, Oct 03 2011
A133314 applied to the derivative of A(x,t) implies (a.+b.)^n = 0^n, for (b_n)=P(n+1,t) and (a_0)=1, (a_1)=-t, and (a_n)=-(1+t) P(n,t) otherwise. E.g., umbrally, (a.+b.)^2 = a_2*b_0 + 2 a_1*b_1 + a_0*b_2 = 0. - Tom Copeland, Oct 08 2011
Beginning with the second column, the rows give the faces of the Whitehouse simplicial complex with the fourth-order complex being three isolated vertices and the fifth-order being the Petersen graph with 10 vertices and 15 edges (cf. Readdy). - Tom Copeland, Oct 03 2014
Stratifications of smooth projective varieties which are fine moduli spaces for stable n-pointed rational curves. Cf. pages 20 and 30 of the Kock and Vainsencher reference and references in A134685. - Tom Copeland, May 18 2017
Named after the American mathematician Morgan Ward (1901-1963). - Amiram Eldar, Jun 26 2021

			Triangle begins:
  1
  1   3
  1  10   15
  1  25  105  105
  1  56  490 1260   945
  1 119 1918 9450 17325 10395
  ...

Louis Comtet, Advanced Combinatorics, Reidel, 1974, page 222.

G. C. Greubel, Rows n = 1..65, flattened
J. Fernando Barbero G., Jesús Salas and Eduardo J. S. Villaseñor, Bivariate Generating Functions for a Class of Linear Recurrences, I: General Structure, Journal of Combinatorial Theory, Series A, Vol. 125 (2014), pp. 146-165; arXiv preprint, arXiv:1307.2010 [math.CO], 2013-2014.
J. Fernando Barbero G., Jesús Salas and Eduardo J. S. Villaseñor, Bivariate Generating Functions for a Class of Linear Recurrences. II. Applications, arXiv preprint arXiv:1307.5624 [math.CO], 2013.
J. Fernando Barbero G., Jesús Salas and Eduardo J. S. Villaseñor, Generalized Stirling permutations and forests: Higher-order Eulerian and Ward numbers, Electronic Journal of Combinatorics, Vol. 22, No. 3 (2015), #P3.37.
Andreas Blass, Natasha Dobrinen and Dilip Raghavan, The next best thing to a p-point, The Journal of Symbolic Logic, Vol. 80, No. 3 (2015), pp. 866-900; arXiv preprint, arXiv:1308.3790 [math.LO], 2013.
David Callan, Toufik Mansour, and Mark Shattuck Some identities for derangement and Ward number sequences and related bijections, Pure Mathematics and Applications, Vol. 25 (Edition 2), pp. 132-143, 2015.
L. Carlitz, The coefficients in an asymptotic expansion and certain related numbers, Duke Math. J., Vol. 35, No. 1 (1968), pp. 83-90.
Lane Clark, Asymptotic normality of the Ward numbers, Discrete Math. 203 (1999), no. 1-3, 41-48. [From _N. J. A. Sloane_, Feb 06 2012]
Tom Copeland, Generators, Inversion, and Matrix, Binomial, and Integral Transforms.
Robert Coquereaux and Jean-Bernard Zuber, Counting partitions by genus. II. A compendium of results, arXiv:2305.01100 [math.CO], 2023. See p. 3.
Bishal Deb and Alan D. Sokal, Higher-order Stirling cycle and subset triangles: Total positivity, continued fractions and real-rootedness, arXiv:2507.18959 [math.CO], 2025. See p. 6.
Satyan L. Devadoss, Daoji Huang and Dominic Spadacene, Polyhedral covers of tree space, SIAM Journal on Discrete Mathematics, Vol. 28, No. 3 (2014), pp. 1508-1514; arXiv preprint, arXiv:1311.0766 [math.CO], 2013.
S. Devadoss and J. Morava, Diagonalizing the genome I: navigation in tree spaces, arXiv:1009.3224v2 [math.AG], 2012.
S. Devadoss and O. Schuh, Cartography of Tree Space, Leonardo (MIT Press Direct), Vol. 53, Issue 3, 2019.
Andreas Dieckmann and Erik Vigren, A New Result in Form of Finite Triple Sums for a Series from Ramanujan's Notebooks, Symmetry (2022) Vol. 14, No. 6, 1090.
Ming-Jian Ding and Jiang Zeng, Proof of an explicit formula for a series from Ramanujan's Notebooks via tree functions, arXiv:2307.00566 [math.CO], 2023.
Brian Drake, Ira M. Gessel and Guoce Xin, Three Proofs and a Generalization of the Goulden-Litsyn-Shevelev Conjecture on a Sequence Arising in Algebraic Geometry, J. of Integer Sequences, Vol. 10 (2007), Article 07.3.7.
Joseph Felsenstein, The number of evolutionary trees, Systematic Biology, 27 (1978), pp. 27-33, 1978.
Giovanni Gaiffi, Nested sets, set partitions and Kirkman-Cayley dissection numbers, arXiv preprint arXiv:1404.3395 [math.CO], 2014.
David M. Jackson, Achim Kempf and Alejandro H. Morales, A robust generalization of the Legendre transform for QFT, Journal of Physics A: Mathematical and Theoretical, Vol. 50, No. 22 (2017), 225201; arXiv preprint, arXiv:1612.0046 [hep-th], 2017.
Bradley Robert Jones, On tree hook length formulae, Feynman rules and B-series, Master's thesis, Math Dept., Simon Fraser University, 2014. p. 23.
Joachim Kock and Israel Vainsencher, Kontsevich's Formula for Rational Plane Curves, Recife, 1999 (version 31/01/2003).
Toufik Mansour and Mark Shattuck, A polynomial generalization of some associated sequences related to set partitions, Periodica Mathematica Hungarica, Vol. 75, No. 2 (December 2017), pp. 398-412.
MathOverflow, Face numbers for tropical Grassmannian G'_2,7 simplicial complex?, 2011.
Andrew Elvey Price and Alan D. Sokal, Phylogenetic trees, augmented perfect matchings, and a Thron-type continued fraction (T-fraction) for the Ward polynomials, arXiv:2001.01468 [math.CO], 2020.
Margaret A. Readdy, The pre-WDVV ring of physics and its topology, The Ramanujan Journal, Vol. 10, No. 2 (2005), pp. 269-281; preprint, 2002.
Lucas Randazzo, Arboretum for a generalisation of Ramanujan polynomials, The Ramanujan Journal, Vol. 54 (2019), pp. 1-14; arXiv preprint, arXiv:1905.02083 [math.CO], 2019.
Lucas Randazzo, Combinatoire bijective autour d'arbres et de chemins, doctoral thesis, Université Paris-Est, 2019.
Peter Regner, Phylogenetic Trees: Selected Combinatorial Problems, Master's Thesis, 2012, Institute of Discrete Mathematics and Geometry, TU Vienna, p. 52.
Fuquan Ren, Jean Yeh, and Roberta Rui Zhou, Context-Free Grammars for Several Triangular Arrays, Axioms, Vol. 11, Issue 6, 297, 2022.
E. G. Santos, Counting non-attacking chess pieces placements: Bishops and Anassas, arXiv:2411.16492 [math.CO], 2024. See p. 8.
Leonard M. Smiley, Completion of a Rational Function Sequence of Carlitz, arXiv:math/0006106 [math.CO], 2000.
Pierre Théorêt, Fonctions génératrices pour une classe d'équations aux différences partielles, Ann. Sci. Math. Quebec 19, 91-105 (1995).
Elena L. Wang and Guoce Xin, On Ward Numbers and Increasing Schröder Trees, arXiv:2507.15654 [math.CO], 2025. See p. 1.
Morgan Ward, The representations of Stirling's numbers and Stirling's polynomials as sums of factorials, Amer. J. Math., Vol. 56, No. 1 (1934), pp. 87-95.
Bao-Xuan Zhu, Coefficientwise Hankel-total positivity of row-generating polynomials for the m-Jacobi-Rogers triangle, arXiv:2202.03793 [math.CO], 2022.

The same as A269939, with column k = 0 removed.
A reshaped version of the triangle of associated Stirling numbers of the second kind, A008299.
A181996 is the mirror image.
Columns k = 2, 3, 4 are A000247, A000478, A058844.
Diagonal k = n is A001147.
Diagonal k = n - 1 is A000457.
Row sums are A000311.
Alternating row sums are signed factorials (-1)^(n-1)*A000142(n).
Cf. A112493.

t[n_, k_] := Sum[(-1)^i*Binomial[n, i]*Sum[(-1)^j*(k-i-j)^(n-i)/(j!*(k-i-j)!), {j, 0, k-i}], {i, 0, k}]; row[n_] := Table[t[k, k-n], {k, n+1, 2*n}]; Table[row[n], {n, 1, 9}] // Flatten (* Jean-François Alcover, Apr 23 2014, after A008299 *)

E.g.f. for the polynomials is A(x,t) = (x-t)/(t+1) + T{ (t/(t+1)) * exp[(x-t)/(t+1)] }, where T(x) is the Tree function, the e.g.f. of A000169. The compositional inverse in x (about x = 0) is B(x) = x + -t * [exp(x) - x - 1]. Special case t = 1 gives e.g.f. for A000311. These results are a special case of A134685 with u(x) = B(x).
From Tom Copeland, Oct 26 2008: (Start)
Umbral-Sheffer formalism gives, for m a positive integer and u = t/(t+1),
[P(.,t)+Q(.,x)]^m = [m Q(m-1,x) - t Q(m,x)]/(t+1) + sum(n>=1) { n^(n-1)[u exp(-u)]^n/n! [n/(t+1)+Q(.,x)]^m }, when the series is convergent for a sequence of functions Q(n,x).
Check: With t=1; Q(n,x)=0^n, for n>=0; and Q(-1,x)=0, then [P(.,1)+Q(.,x)]^m = P(m,1) = A000311(m).
(End)
Let h(x,t) = 1/(dB(x)/dx) = 1/(1-t*(exp(x)-1)), an e.g.f. in x for row polynomials in t of A019538, then the n-th row polynomial in t of the table A134991, P(n,t), is given by ((h(x,t)*d/dx)^n)x evaluated at x=0, i.e., A(x,t) = exp(x*P(.,t)) = exp(x*h(u,t)*d/du) u evaluated at u=0. Also, dA(x,t)/dx = h(A(x,t),t). - Tom Copeland, Sep 05 2011
The polynomials (1+t)/t*P(n,t) are the row polynomials of A112493. Let f(x) = (1+x)/(1-x*t). Then for n >= 0, P(n+1,t) is given by t/(1+t)*(f(x)*d/dx)^n(f(x)) evaluated at x = 0. - Peter Bala, Sep 30 2011
From Tom Copeland, Oct 04 2011: (Start)
T(n,k) = (k+1)*T(n-1,k) + (n+k+1)*T(n-1,k-1) with starting indices n=0 and k=0 beginning with P(2,t) (as suggested by a formula of David Speyer on MathOverflow).
T(n,k) = k*T(n-1,k) + (n+k-1)*T(n-1,k-1) with starting indices n=1 and k=1 of table (cf. Smiley above and Riordin ref.[10] therein).
P(n,t) = (1/(1+t))^n * Sum_{k>=1} k^(n+k-1)*(u*exp(-u))^k / k! with u=(t/(t+1)) for n>1; therefore, Sum_{k>=1} (-1)^k k^(n+k-1) x^k/k! = [1+LW(x)]^(-n) P{n,-LW(x)/[1+LW(x)]}, with LW(x) the Lambert W-Fct.
T(n,k) = Sum_{i=0..k} ((-1)^i binomial(n+k,i) Sum_{j=0..k-i} (-1)^j (k-i-j)^(n+k-i)/(j!(k-i-j)!)) from relation to A008299. (End)
The e.g.f. A(x,t) = -v * ( Sum_{j=>1} D(j-1,u) (-z)^j / j! ) where u = (x-t)/(1+t), v = 1+u, z = x/((1+t) v^2) and D(j-1,u) are the polynomials of A042977. dA/dx = 1/((1+t)(v-A)) = 1/(1-t*(exp(A)-1)). - Tom Copeland, Oct 06 2011
The general results on the convolution of the refined partition polynomials of A134685, with u_1 = 1 and u_n = -t otherwise, can be applied here to obtain results of convolutions of these polynomials. - Tom Copeland, Sep 20 2016
E.g.f.: C(u,t) = (u-t)/(1+t) - W( -((t*exp((u-t)/(1+t)))/(1+t)) ), where W is the principal value of the Lambert W-function. - Cheng Peng, Sep 11 2021
The function C(u,t) in the previous formula by Peng is precisely the function A(u,t) given in the initial 2008 formula of this section and the Oct 06 2011 formula from Copeland. As noted in A000169, Euler's tree function is T(x) = -LambertW(-x), where W(x) is the principal branch of Lambert's function, and T(x) is the e.g.f. of A000169. - Tom Copeland, May 13 2022

Reference to A181996 added by N. J. A. Sloane, Apr 05 2012

Further edits by N. J. A. Sloane, Jan 24 2020

A368927 Number of labeled loop-graphs covering a subset of {1..n} such that it is possible to choose a different vertex from each edge.

Original entry on oeis.org

1, 2, 7, 39, 314, 3374, 45630, 744917, 14245978, 312182262, 7708544246, 211688132465, 6397720048692, 210975024924386, 7537162523676076, 289952739051570639, 11949100971787370300, 525142845422124145682, 24515591201199758681892, 1211486045654016217202663

Offset: 0

Gus Wiseman, Jan 15 2024

nonn

These are loop-graphs where every connected component has a number of edges less than or equal to the number of vertices. Also loop-graphs with at most one cycle (unicyclic) in each connected component.

			The a(0) = 1 through a(2) = 7 loop-graphs (loops shown as singletons):
  {}  {}     {}
      {{1}}  {{1}}
             {{2}}
             {{1,2}}
             {{1},{2}}
             {{1},{1,2}}
             {{2},{1,2}}

Andrew Howroyd, Table of n, a(n) for n = 0..200

Without the choice condition we have A006125.
The case of a unique choice is A088957, unlabeled A087803.
The case without loops is A133686, complement A367867, covering A367869.
For exactly n edges and no loops we have A137916, unlabeled A137917.
For exactly n edges we have A333331 (maybe), complement A368596.
For edges of any positive size we have A367902, complement A367903.
The covering case is A369140, complement A369142.
The complement is counted by A369141.
The complement for n edges and no loops is A369143, covering A369144.
The unlabeled version is A369145, complement A369146.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A006129 counts covering graphs, unlabeled A002494.
A322661 counts labeled covering loop-graphs, connected A062740.
Cf. A000169, A000272, A000666, A014068, A057500, A116508, A367863, A368601, A368924, A368984, A369200.

Table[Length[Select[Subsets[Subsets[Range[n],{1,2}]], Length[Select[Tuples[#],UnsameQ@@#&]]!=0&]],{n,0,5}]

seq(n)={my(t=-lambertw(-x + O(x*x^n))); Vec(serlaplace(exp(3*t/2 - 3*t^2/4)/sqrt(1-t) ))} \\ Andrew Howroyd, Feb 02 2024

Binomial transform of A369140.
Exponential transform of A369197 with A369197(1) = 2.
E.g.f.: exp(3*T(x)/2 - 3*T(x)^2/4)/sqrt(1-T(x)), where T(x) is the e.g.f. of A000169. - Andrew Howroyd, Feb 02 2024

a(7) onwards from Andrew Howroyd, Feb 02 2024

A333331 Number of integer points in the convex hull in R^n of parking functions of length n.

Original entry on oeis.org

1, 3, 17, 144, 1623, 22804, 383415, 7501422

Offset: 1

Richard Stanley, Mar 15 2020

It is observed by Gus Wiseman in A368596 and A368730 that this sequence appears to be the complement of those sequences. If this is the case, then a(n) is the number of labeled graphs with loops allowed in which each connected component has an equal number of vertices and edges and the conjectured formula holds. Terms for n >= 9 are expected to be 167341283, 4191140394, 116425416531, ... - Andrew Howroyd, Jan 10 2024
From Gus Wiseman, Mar 22 2024: (Start)
An equivalent conjecture is that a(n) is the number of loop-graphs with n vertices and n edges such that it is possible to choose a different vertex from each edge. I call these graphs choosable. For example, the a(3) = 17 choosable loop-graphs are the following (loops shown as singletons):
  {{1},{2},{3}}  {{1},{2},{1,3}}  {{1},{1,2},{1,3}}  {{1,2},{1,3},{2,3}}
                 {{1},{2},{2,3}}  {{1},{1,2},{2,3}}
                 {{1},{3},{1,2}}  {{1},{1,3},{2,3}}
                 {{1},{3},{2,3}}  {{2},{1,2},{1,3}}
                 {{2},{3},{1,2}}  {{2},{1,2},{2,3}}
                 {{2},{3},{1,3}}  {{2},{1,3},{2,3}}
                                  {{3},{1,2},{1,3}}
                                  {{3},{1,2},{2,3}}
                                  {{3},{1,3},{2,3}}
(End)

			For n=2 the parking functions are (1,1), (1,2), (2,1). They are the only integer points in their convex hull. For n=3, in addition to the 16 parking functions, there is the additional point (2,2,2).

R. P. Stanley (Proposer), Problem 12191, Amer. Math. Monthly, 127:6 (2020), 563.

Thomas Selig, The stochastic sandpile model on complete graphs, arXiv:2209.07301 [math.PR], 2022.

All of the following relative references pertain to the conjecture:
The case of unique choice A000272.
The version without the choice condition is A014068, covering A368597.
The case of just pairs A137916.
For any number of edges of any positive size we have A367902.
The complement A368596, covering A368730.
Allowing edges of any positive size gives A368601, complement A368600.
Counting by singletons gives A368924.
For any number of edges we have A368927, complement A369141.
The connected case is A368951.
The unlabeled version is A368984, complement A368835.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A006125 counts graphs, unlabeled A000088.
A058891 counts set-systems (without singletons A016031), unlabeled A000612.
Cf. A000169, A000666, A057500, A062740, A129271, A133686, A367863, A367903, A369146, A369199.

Conjectured e.g.f.: exp(-log(1-T(x))/2 + T(x)/2 - T(x)^2/4) where T(x) = -LambertW(-x) is the e.g.f. of A000169. - Andrew Howroyd, Jan 10 2024

cp's OEIS Frontend

A055302 Triangle of number of labeled rooted trees with n nodes and k leaves, n >= 1, 1 <= k <= n.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A063170 Schenker sums with n-th term.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

PARI

Python

UBASIC

Formula

A003992 Square array read by upwards antidiagonals: T(n,k) = n^k for n >= 0, k >= 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Maxima

PARI

Formula

A052182 Determinant of n X n matrix whose rows are cyclic permutations of 1..n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

MuPAD

PARI

Formula

Extensions

A052750 a(n) = (2*n + 1)^(n - 1).

Views

Author

Keywords

Comments