A052750 -id:A052750 - cp's OEIS frontend

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

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

Offset: 0

N. J. A. Sloane

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

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

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Anders Björner and Francesco Brenti, Combinatorics of Coxeter groups. Graduate Texts in Mathematics, 231. Springer, New York, 2005.
Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 311.
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Seiichi Manyama, Table of n, a(n) for n = 0..388 (terms 0..100 from N. J. A. Sloane)
Federico Ardila, Matthias Beck, and Jodi McWhirter, The Arithmetic of Coxeter Permutahedra, arXiv:2004.02952 [math.CO], 2020.
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M. Beck, A. Berrizbeitia, M. Dairyko, C. Rodriguez, A. Ruiz, and S. Veeneman., Parking functions, Shi arrangements, and mixed graphs, Amer. Math. Monthly, 122 (2015), 660-673.
Norman L. Biggs, Chip-firing and the critical group of a graph, J. Algeb. Combin., 9 (1999), 25-45.
Norman L. Biggs, E. Keith Lloyd, and Robin J. Wilson, Graph Theory 1736-1936, Oxford, 1976, p. 51.
Richard P. Brent, M. L. Glasser, and Anthony J. Guttmann, A Conjectured Integer Sequence Arising From the Exponential Integral, arXiv:1812.00316 [math.NT], 2018.
Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.
David Callan, A Combinatorial Derivation of the Number of Labeled Forests, J. Integer Seqs., Vol. 6, 2003.
Saverio Caminiti and Emanuele G. Fusco, On the Number of Labeled k-arch Graphs, Journal of Integer Sequences, Vol 10 (2007), Article 07.7.5
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Huantian Cao, AutoGF: An Automated System to Calculate Coefficients of Generating Functions. [Broken link]
R. Castelo and A. Siebes, A characterization of moral transitive directed acyclic graph Markov models as labeled trees, Report CS-2000-44, Department of Computer Science, Univ. Utrecht.
R. Castelo and A. Siebes, A characterization of moral transitive acyclic directed graph Markov models as labeled trees, J. Statist. Planning Inference, 115(1):235-259, 2003.
A. Cayley, A theorem on trees, Quart. J. Pure and Applied Math., 23 (1889), 376-378.
Jacob W. Cooper, Adam Kabela, Daniel Král', and Théo Pierron, Hadwiger meets Cayley, arXiv:2005.05989 [math.CO], 2020.
S. Coulomb and M. Bauer, On vertex covers, matchings and random trees, arXiv:math/0407456 [math.CO], 2004.
FindStat - Combinatorial Statistic Finder, Parking functions
J. Gilbey and L. Kalikow, Parking functions, valet functions and priority queues, Discrete Math., 197 (1999), 351-375.
M. Golin and S. Zaks, Labeled trees and pairs of input-output permutations in priority queues, Theoret. Comput. Sci., 205 (1998), 99-114.
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F. A. Haight, Letter to N. J. A. Sloane, n.d.
A. M. Hamel, Priority queue sorting and labeled trees, Annals Combin., 7 (2003), 49-54.
Angela Hicks, Combinatorics of the Diagonal Harmonics, in Recent Trends in Algebraic Combinatorics, part of the Association for Women in Mathematics Series (2019) Vol 16. Springer, Cham, 159-188.
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D. M. Jackson, Some Combinatorial Problems Associated with Products of Conjugacy Classes of the Symmetric Group, Journal of Combinatorial Theory, Series A, 49 363-369(1988).
S. Janson, D. E. Knuth, T. Łuczak and B. Pittel, The Birth of the Giant Component, Random Structures and Algorithms Vol. 4 (1993), 233-358.
C. Ji and J. Propp, Brussels Sprouts, Noncrossing Trees, and Parking Functions, arXiv preprint arXiv:1805.03608 [math.CO], 2018.
L. Kalikow, Symmetries in trees and parking functions, Discrete Math., 256 (2002), 719-741.
C. Lamathe, The Number of Labelled k-Arch Graphs, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.1.
A. Laradji and A. Umar, On the number of nilpotents in the partial symmetric semigroup, Comm. Algebra 32 (2004), 3017-3023. From _Abdullahi Umar_, Aug 25 2008
C. J. Liu and Yutze Chow, On operator and formal sum methods for graph enumeration problems, SIAM J. Algebraic Discrete Methods, 5 (1984), no. 3, 384--406. MR0752043 (86d:05059). See Eq. (47). - From _N. J. A. Sloane_, Apr 09 2014
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J. Pitman, Coalescent Random Forests, J. Combin. Theory, A85 (1999), 165-193.
J.-B. Priez and A. Virmaux, Non-commutative Frobenius characteristic of generalized parking functions: Application to enumeration, arXiv preprint arXiv:1411.4161 [math.CO], 2014.
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Index entries for sequences related to trees
Index entries for "core" sequences

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

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

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

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

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

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

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

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

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

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

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

A052752 a(n) = (3*n+1)^(n-1).

Original entry on oeis.org

1, 1, 7, 100, 2197, 65536, 2476099, 113379904, 6103515625, 377801998336, 26439622160671, 2064377754059776, 177917621779460413, 16777216000000000000, 1718264124282290785243, 189937030341242876870656, 22539340290692258087863249, 2857942574656970690381479936

Offset: 0

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

G. C. Greubel, Table of n, a(n) for n = 0..333
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 708
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.

Cf. A000169, A052750, A052774, A052782, A000272, A001711.

[(3*n+1)^(n-1): n in [0..50]]; // G. C. Greubel, Nov 16 2017

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

Table[(3n+1)^(n-1),{n,0,20}] (* Harvey P. Dale, Aug 14 2015 *)
With[{nmax = 50}, CoefficientList[Series[Exp[-LambertW[-3*x]/3], {x, 0, nmax}], x]*Range[0, nmax]!] (* G. C. Greubel, Nov 16 2017 *)

for(n=0,50, print1((3*n+1)^(n-1), ", ")) \\ G. C. Greubel, Nov 16 2017

x='x+O('x^50); Vec(serlaplace(exp(-lambertw(-3*x)/3))) \\ G. C. Greubel, Nov 16 2017

E.g.f.: exp(-(1/3)*LambertW(-3*x)).
From Peter Bala, Dec 19 2013: (Start)
The e.g.f. A(x) = 1 + x + 7*x^2/2! + 100*x^3/3! + 2197*x^4/4! + ... satisfies:
1) A(x*exp(-3*x)) = exp(x) = 1/A(-x*exp(3*x));
2) A^3(x) = 1/x*series reversion(x*exp(-3*x));
3) A(x^3) = 1/x*series reversion(x*exp(-x^3));
4) A(x) = exp(x*A(x)^3);
5) A(x) = 1/A(-x*A(x)^6). (End)
E.g.f.: (-LambertW(-3*x)/(3*x))^(1/3). - Vaclav Kotesovec, Dec 07 2014
Related to A001711 by Sum_{n >= 1} a(n)*x^n/n! = series reversion( 1/(1 + x)^3*log(1 + x) ) = series reversion(x - 7*x^2/2! + 47*x^3/3! - 342*x^4/4! + ...). Cf. A000272, A052750. - Peter Bala, Jun 15 2016
a(n) = Sum_{k=1..n} (-1)^(n-k)*(2n+k)^(n-1)*binomial(n,k-1), a(0)=1. - Vladimir Kruchinin, Aug 14 2025

Better description from Vladeta Jovovic, Sep 02 2003

A085527 a(n) = (2n+1)^n.

Original entry on oeis.org

1, 3, 25, 343, 6561, 161051, 4826809, 170859375, 6975757441, 322687697779, 16679880978201, 952809757913927, 59604644775390625, 4052555153018976267, 297558232675799463481, 23465261991844685929951, 1977985201462558877934081, 177482997121587371826171875

Offset: 0

N. J. A. Sloane, Jul 05 2003

a(n) is the determinant of the zigzag matrix Z(n) (see A088961). - Paul Boddington, Nov 03 2003
a(n) is also the number of rho-labeled graphs with n edges. A graph with n edges is a rho-labeled graph if there exists a one-to-one mapping from its vertex set to {0,1,...,2n} such that every edge receives as a label the absolute difference of its end-vertices and the edge labels are x1,x2,...,xn where xi=i or xi=2n+1-i. - Christian Barrientos and Sarah Minion, Feb 20 2015
a(n) is the number of nodes in the canonical automaton for the affine Weyl group of types B_n and C_n. - Tom Edgar, May 12 2016
a(n) is the number of rooted (at an edge) 2-trees with n+2 edges. See also A052750. - Nikos Apostolakis, Dec 05 2018

Anders Björner and Francesco Brenti, Combinatorics of Coxeter groups. Graduate Texts in Mathematics, 231. Springer, New York, 2005.

G. C. Greubel, Table of n, a(n) for n = 0..350
Karola Mészáros, Labeling the Regions of the Type C_n Shi Arrangement, The Electronic Journal of Combinatorics, vol. 20, no. 2, (2013).
Zhi-Wei Sun, Fedor Petrov, A surprising identity, MathOverflow, Jan 17 2019.

Cf. A062971, A085528, A088961, A099753, A214406, A052750.

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

[(2*n+1)^n: n in [0..20]]; // Wesley Ivan Hurt, Mar 01 2015

A085527:=n->(2*n+1)^n: seq(A085527(n), n=0..20); # Wesley Ivan Hurt, Mar 01 2015

Table[(2 n + 1)^n, {n, 0, 20}] (* Wesley Ivan Hurt, Mar 01 2015 *)

PARI
```
a(n)=(2*n+1)^n;
```

def A085527(n): return ((n<<1)|1)**n # Chai Wah Wu, Nov 10 2024

E.g.f.: sqrt(2)/(2*(1+LambertW(-2*x))*sqrt(-x/LambertW(-2*x))). - Vladeta Jovovic, Oct 16 2004
For r = 0, 1, 2, ..., the e.g.f. for the sequence whose n-th term is (2*n+1)^(n+r) can be expressed in terms of the function U(z) = Sum_{n >= 0} (2*n+1)^(n-1)*z^(2*n+1)/(2^n*n!). See A214406 for details. In the present case, r = 0, and the resulting e.g.f. is 1/z*U(z)/(1 - U(z)^2) taken at z = sqrt(2*x). - Peter Bala, Aug 06 2012
a(n) = [x^n] 1/(1 - (2*n+1)*x). - Ilya Gutkovskiy, Oct 10 2017
a(n) = (-2)^n * D(2*n + 1), where D(n) is the determinant of the n X n matrix M with elements M(j, k) = cos(Pi*j*k/n). See the Zhi-Wei Sun, Petrov link. - Peter Luschny, Sep 19 2021
a(n) ~ exp(1/2) * 2^n * n^n. - Vaclav Kotesovec, Dec 05 2021
Series reversion of (1 - x)^2 * log(1/(1 - x)) begins  x + 3*x^2/2! + 25*x^3/3! + 343*x^4/4! + 6561*x^5/5! + .... - Peter Bala, Sep 27 2023
a(n) = Product_{k=1..n} tan(k*Pi/(1+2*n))^(2*n). - Chai Wah Wu, Nov 10 2024

A130777 Coefficients of first difference of Chebyshev S polynomials.

Original entry on oeis.org

1, -1, 1, -1, -1, 1, 1, -2, -1, 1, 1, 2, -3, -1, 1, -1, 3, 3, -4, -1, 1, -1, -3, 6, 4, -5, -1, 1, 1, -4, -6, 10, 5, -6, -1, 1, 1, 4, -10, -10, 15, 6, -7, -1, 1, -1, 5, 10, -20, -15, 21, 7, -8, -1, 1, -1, -5, 15, 20, -35, -21, 28, 8, -9, -1, 1, 1, -6, -15, 35, 35, -56, -28, 36, 9, -10, -1, 1

Offset: 0

Philippe Deléham, Jul 14 2007

Inverse of triangle in A061554.
Signed version of A046854.
From Paul Barry, May 21 2009: (Start)
Riordan array ((1-x)/(1+x^2),x/(1+x^2)).
This triangle is the coefficient triangle for the Hankel transforms of the family of generalized Catalan numbers that satisfy a(n;r)=r*a(n-1;r)+sum{k=1..n-2, a(k)*a(n-1-k;r)}, a(0;r)=a(1;r)=1. The Hankel transform of a(n;r) is h(n)=sum{k=0..n, T(n,k)*r^k} with g.f. (1-x)/(1-r*x+x^2). These sequences include A086246, A000108, A002212. (End)
From Wolfdieter Lang, Jun 11 2011: (Start)
The Riordan array ((1+x)/(1+x^2),x/(1+x^2)) with entries Phat(n,k)= ((-1)^(n-k))*T(n,k) and o.g.f. Phat(x,z)=(1+z)/(1-x*z+z^2) for the row polynomials Phat(n,x) is related to Chebyshev C and S polynomials as follows.
Phat(n,x) = (R(n+1,x)-R(n,x))/(x+2) = S(2*n,sqrt(2+x))
  with R(n,x)=C_n(x) in the Abramowitz and Stegun notation, p. 778, 22.5.11. See A049310 for the S polynomials. Proof from the o.g.f.s.
Recurrence for the row polynomials Phat(n,x):
  Phat(n,x) = x*Phat(n-1,x) - Phat(n-2,x) for n>=1; Phat(-1,x)=-1, Phat(0,x)=1.
The A-sequence for this Riordan array Phat (see the W. Lang link under A006232 for A- and Z-sequences for Riordan matrices) is given by 1, 0, -1, 0, -1, 0, -2, 0, -5,.., starting with 1 and interlacing the negated A000108 with zeros (o.g.f. 1/c(x^2) = 1-c(x^2)*x^2, with the o.g.f. c(x) of A000108).
The Z-sequence has o.g.f. sqrt((1-2*x)/(1+2*x)), and it is given by A063886(n)*(-1)^n.
The A-sequence of the Riordan array T(n,k) is identical with the one for the Riordan array Phat, and the Z-sequence is -A063886(n).
(End)
The row polynomials P(n,x) are the characteristic polynomials of the adjacency matrices of the graphs which look like P_n (n vertices (nodes), n-1 lines (edges)), but vertex no. 1 has a loop. - Wolfdieter Lang, Nov 17 2011
From Wolfdieter Lang, Dec 14 2013: (Start)
The zeros of P(n,x) are x(n,j) = -2*cos(2*Pi*j/(2*n+1)), j=1..n. From P(n,x) = (-1)^n*S(2*n,sqrt(2-x)) (see, e.g., the Lemma 6 of the W. Lang link).
The discriminants of the P-polynomials are given in A052750. (End)

			The triangle T(n,k) begins:
n\k  0   1   1   3    4    5    6    7    8    9  10  11  12  13 14 15 ...
0:   1
1:  -1   1
2:  -1  -1   1
3:   1  -2  -1   1
4:   1   2  -3  -1    1
5:  -1   3   3  -4   -1    1
6:  -1  -3   6   4   -5   -1    1
7:   1  -4  -6  10    5   -6   -1    1
8:   1   4 -10 -10   15    6   -7   -1    1
9:  -1   5  10 -20  -15   21    7   -8   -1    1
10: -1  -5  15  20  -35  -21   28    8   -9   -1   1
11:  1  -6 -15  35   35  -56  -28   36    9  -10  -1   1
12:  1   6 -21 -35   70   56  -84  -36   45   10 -11  -1   1
13: -1   7  21 -56  -70  126   84 -120  -45   55  11 -12  -1   1
14: -1  -7  28  56 -126 -126  210  120 -165  -55  66  12 -13  -1  1
15:  1  -8 -28  84  126 -252 -210  330  165 -220 -66  78  13 -14 -1  1
...  reformatted and extended - _Wolfdieter Lang_, Jul 31 2014.
---------------------------------------------------------------------------
From _Paul Barry_, May 21 2009: (Start)
Production matrix is
-1, 1,
-2, 0, 1,
-2, -1, 0, 1,
-4, 0, -1, 0, 1,
-6, -1, 0, -1, 0, 1,
-12, 0, -1, 0, -1, 0, 1,
-20, -2, 0, -1, 0, -1, 0, 1,
-40, 0, -2, 0, -1, 0, -1, 0, 1,
-70, -5, 0, -2, 0, -1, 0, -1, 0, 1 (End)
Row polynomials as first difference of S polynomials:
P(3,x) = S(3,x) - S(2,x) = (x^3 - 2*x) - (x^2 -1) = 1 - 2*x - x^2 +x^3.
Alternative triangle recurrence (see a comment above): T(6,2) = T(5,2) + T(5,1) = 3 + 3 = 6. T(6,3) = -T(5,3) + 0*T(5,1) = -(-4) = 4. - _Wolfdieter Lang_, Jul 31 2014

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964. Tenth printing, Wiley, 2002 (also electronically available).

Hyeong-Kwan Ju, On the sequence generated by a certain type of matrices, Honam Math. J. 39, No. 4, 665-675 (2017), Theorem 2.16.
Wolfdieter Lang, The field Q(2cos(pi/n)), its Galois group and length ratios in the regular n-gon, arXiv:1210.1018 [math.GR], 2012-2017; see Definition 1, Lemma 6 and Remark 4.
P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), p. 22-31 (formula 5).
Index entries for sequences related to Chebyshev polynomials.

Cf. A066170, A046854, A057077 (first column).

Row sums: A010892(n+1); repeat(1,0,-1,-1,0,1). Alternating row sums: A061347(n+2); repeat(1,-2,1).

A130777 := proc(n,k): (-1)^binomial(n-k+1,2)*binomial(floor((n+k)/2),k) end: seq(seq(A130777(n,k), k=0..n), n=0..11); # Johannes W. Meijer, Aug 08 2011

T[n_, k_] := (-1)^Binomial[n - k + 1, 2]*Binomial[Floor[(n + k)/2], k];
Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 14 2017, from Maple *)

@CachedFunction
def A130777(n,k):
    if n< 0: return 0
    if n==0: return 1 if k == 0 else 0
    h = A130777(n-1,k) if n==1 else 0
    return A130777(n-1,k-1) - A130777(n-2,k) - h
for n in (0..9): [A130777(n,k) for k in (0..n)] # Peter Luschny, Nov 20 2012

Number triangle T(n,k) = (-1)^C(n-k+1,2)*C(floor((n+k)/2),k). - Paul Barry, May 21 2009
From Wolfdieter Lang, Jun 11 2011: (Start)
Row polynomials: P(n,x) = sum(k=0..n, T(n,k)*x^k) = R(2*n+1,sqrt(2+x)) / sqrt(2+x), with Chebyshev polynomials R with coefficients given in A127672 (scaled T-polynomials).
R(n,x) is called C_n(x) in Abramowitz and Stegun's handbook, p. 778, 22.5.11.
P(n,x) = S(n,x)-S(n-1,x), n>=0, S(-1,x)=0, with the Chebyshev S-polynomials (see the coefficient triangle A049310).
O.g.f. for row polynomials: P(x,z):= sum(n>=0, P(n,x)*z^n ) = (1-z)/(1-x*z+z^2).
  (from the o.g.f. for R(2*n+1,x), n>=0, computed from the o.g.f. for the R-polynomials (2-x*z)/(1-x*z+z^2) (see A127672))
Proof of the Chebyshev connection from the o.g.f. for Riordan array property of this triangle (see the P. Barry comment above).
For the A- and Z-sequences of this Riordan array see a comment above. (End)
abs(T(n,k)) = A046854(n,k) = abs(A066170(n,k)) T(n,n-k) = A108299(n,k); abs(T(n,n-k)) = A065941(n,k). - Johannes W. Meijer, Aug 08 2011
From Wolfdieter Lang, Jul 31 2014: (Start)
Similar to the triangles A157751, A244419 and A180070 one can give for the row polynomials P(n,x) besides the usual three term recurrence another one needing only one recurrence step. This uses also a negative argument, namely P(n,x) = (-1)^(n-1)*(-1 + x/2)*P(n-1,-x) + (x/2)*P(n-1,x), n >= 1, P(0,x) = 1. Proof by computing the o.g.f. and comparing with the known one. This entails the alternative triangle recurrence T(n,k) = (-1)^(n-k)*T(n-1,k) + (1/2)*(1 + (-1)^(n-k))*T(n-1,k-1), n >= m >= 1, T(n,k) = 0 if n < k and T(n,0) = (-1)^floor((n+1)/2) = A057077(n+1). [P(n,x) recurrence corrected Aug 03 2014]
(End)

New name and Chebyshev comments by Wolfdieter Lang, Jun 11 2010

A145905 Square array read by antidiagonals: Hilbert transform of triangle A060187.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 9, 5, 1, 1, 27, 25, 7, 1, 1, 81, 125, 49, 9, 1, 1, 243, 625, 343, 81, 11, 1, 1, 729, 3125, 2401, 729, 121, 13, 1, 1, 2187, 15625, 16807, 6561, 1331, 169, 15, 1, 1, 6561, 78125, 117649, 59049, 14641, 2197, 225, 17, 1, 1, 19683, 390625, 823543

Offset: 0

Peter Bala, Oct 27 2008

Definition of the Hilbert transform of a triangular array:
For many square arrays in the database the entries in a row are polynomial in the column index, of degree d say and hence the row generating function has the form P(x)/(1-x)^(d+1), where P is some polynomial function. Often the array whose rows are formed from the coefficients of these P polynomials is of independent interest. This suggests the following definition.
Let [L(n,k)]n,k>=0 be a lower triangular array and let R(n,x) := sum {k = 0 .. n} L(n,k)*x^k, denote the n-th row generating polynomial of L. Then we define the Hilbert transform of L, denoted Hilb(L), to be the square array whose n-th row, n >= 0, has the generating function R(n,x)/(1-x)^(n+1).
In this particular case, L is the array A060187, the array of Eulerian numbers of type B, whose row polynomials are the h-polynomials for permutohedra of type B. The Hilbert transform is an infinite Vandermonde matrix V(1,3,5,...).
We illustrate the Hilbert transform with a few examples:
(1) The Delannoy number array A008288 is the Hilbert transform of Pascal's triangle A007318 (view as the array of coefficients of h-polynomials of n-dimensional cross polytopes).
(2) The transpose of the array of nexus numbers A047969 is the Hilbert transform of the triangle of Eulerian numbers A008292 (best viewed in this context as the coefficients of h-polynomials of n-dimensional permutohedra of type A).
(3) The sequence of Eulerian polynomials begins [1, x, x + x^2, x + 4*x^2 + x^3, ...]. The coefficients of these polynomials are recorded in triangle A123125, whose Hilbert transform is A004248 read as square array.
(4) A108625, the array of crystal ball sequences for the A_n lattices, is the Hilbert transform of A008459 (viewed as the triangle of coefficients of h-polynomials of n-dimensional associahedra of type B).
(5) A142992, the array of crystal ball sequences for the C_n lattices, is the Hilbert transform of A086645, the array of h-vectors for type C root polytopes.
(6) A108553, the array of crystal ball sequences for the D_n lattices, is the Hilbert transform of A108558, the array of h-vectors for type D root polytopes.
(7) A086764, read as a square array, is the Hilbert transform of the rencontres numbers A008290.
(8) A143409 is the Hilbert transform of triangle A073107.

			Triangle A060187 (with an offset of 0) begins
1;
1, 1;
1, 6, 1;
so the entries in the first three rows of the Hilbert transform of
A060187 come from the expansions:
Row 0: 1/(1-x) = 1 + x + x^2 + x^3 + ...;
Row 1: (1+x)/(1-x)^2 = 1 + 3*x + 5*x^2 + 7*x^3 + ...;
Row 2: (1+6*x+x^2)/(1-x)^3 = 1 + 9*x + 25*x^2 + 49*x^3 + ...;
The array begins
n\k|..0....1.....2.....3......4
================================
0..|..1....1.....1.....1......1
1..|..1....3.....5.....7......9
2..|..1....9....25....49.....81
3..|..1...27...125...343....729
4..|..1...81...625..2401...6561
5..|..1..243..3125.16807..59049
...

Ghislain R. Franssens, On a Number Pyramid Related to the Binomial, Deleham, Eulerian, MacMahon and Stirling number triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.4.1.
S. Parker, The Combinatorics of Functional Composition and Inversion, Ph.D. Dissertation, Brandeis Univ. (1993) [From _Tom Copeland_, Nov 09 2008]

Cf. A008292, A039755, A052750 (first superdiagonal), A060187, A114172, A145901.

T:=(n,k) -> (2*k + 1)^n: seq(seq(T(n-k,k),k = 0..n),n = 0..10);

T(n,k) = (2*k + 1)^n, (see equation 4.10 in [Franssens]). This array is the infinite Vandermonde matrix V(1,3,5,7, ....) having a LDU factorization equal to A039755 * diag(2^n*n!) * transpose(A007318).

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A384718 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of e.g.f. B(x)^k, where B(x) is the e.g.f. of A052750.

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

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A052752 a(n) = (3*n+1)^(n-1).

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A085527 a(n) = (2n+1)^n.

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A130777 Coefficients of first difference of Chebyshev S polynomials.

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A145905 Square array read by antidiagonals: Hilbert transform of triangle A060187.

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