A052752 -id:A052752 - cp's OEIS frontend

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

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

A274265 a(n) = (3*n - 1)^(n-1).

Original entry on oeis.org

1, 5, 64, 1331, 38416, 1419857, 64000000, 3404825447, 208827064576, 14507145975869, 1125899906842624, 96549157373046875, 9065737908494995456, 925103102315013629321, 101938319743841411792896, 12063348350820368238715343, 1525878906250000000000000000

Offset: 1

Peter Bala, Jun 19 2016

Compare with A052752.

Cf. A000169, A052752, A085527, A274266, A274267, A274269.

[(3*n-1)^(n-1): n in [1..20]]; // Vincenzo Librandi, Jun 20 2016

A274265 := n -> (3*n - 1)^(n-1):
seq(A274265(n), n = 1..20);

Table[(3*n-1)^(n-1), {n,1,25}] (* G. C. Greubel, Jun 19 2016 *)

E.g.f. A(x) = 1 - exp(-1/3*T(3*x)) = x + 5*x^2/2! + 8^2*x^3/3! + 11^3*x^4/4! + 14^4*x^5/5! + ..., where T(x) = Sum_{n >= 1} n^(n-1)*x^n/n! is Euler's tree function - see A000169.
A(x) = series reversion( (1 - x)^3*log(1/(1 - x)) ). See A274266.
1 - A(x) = exp(-x/(1 - A(x))^3) = exp(-x/(exp(-3*x/(exp(-3*x/ ...))))).
1 - A(-x*exp(3*x)) = exp(x) = 1/(1 - A(x*exp(-3*x))).
1/(1 - A(x)) = Sum_{n >= 0} (3*n + 1)^(n-1)*x^n/n!, the e.g.f. for A052752.

A052774 a(n) = (4*n+1)^(n-1).

Original entry on oeis.org

1, 1, 9, 169, 4913, 194481, 9765625, 594823321, 42618442977, 3512479453921, 327381934393961, 34050628916015625, 3909821048582988049, 491258904256726154641, 67046038752496061076057, 9876832533361318095112441, 1562069488955406402587890625

Offset: 0

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

G. C. Greubel, Table of n, a(n) for n = 0..320
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 731
J.-C. Novelli, J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv:1403.5962 [math.CO], 2014.

Cf. A000169, A052750, A052752, A052782, A000272, A001716.

[(4*n+1)^(n-1): n in [0..30]]; // G. C. Greubel, Nov 14 2017

spec := [S,{B=Prod(Z,S,S,S,S),S=Set(B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
A052774:=n->(4*n+1)^(n-1); seq(A052774(n), n=0..20); # Wesley Ivan Hurt, Dec 03 2013

Table[(4n+1)^(n-1), {n,0,20}] (* Wesley Ivan Hurt, Dec 03 2013 *)
With[{nmax = 50}, CoefficientList[Series[Exp[-LambertW[-4*x]/4], {x, 0, nmax}], x]*Range[0, nmax]!] (* G. C. Greubel, Nov 14 2017 *)

for(n=0, 30, print1((4*n+1)^(n-1), ", ")) \\ G. C. Greubel, Nov 14 2017

x='x+O('x^50); Vec(serlaplace(exp(-lambertw(-4*x)/4))) \\ G. C. Greubel, Nov 14 2017

E.g.f.: exp(-1/4*LambertW(-4*x)).
a(n) = A016813(n)^A023443(n). - Wesley Ivan Hurt, Dec 03 2013
From Peter Bala, Dec 19 2013: (Start)
The e.g.f. A(x) = 1 + x + 9*x^2/2! + 169*x^3/3! + 4913*x^4/4! + ... satisfies:
1) A(x*exp(-4*x)) = exp(x) = 1/A(-x*exp(4*x));
2) A^4(x) = 1/x*series reversion(x*exp(-4*x));
3) A(x^4) = 1/x*series reversion(x*exp(-x^4));
4) A(x) = exp(x*A(x)^4);
5) A(x) = 1/A(-x*A(x)^8). (End)
E.g.f.: (-LambertW(-4*x)/(4*x))^(1/4). - Vaclav Kotesovec, Dec 07 2014
Related to A001716 by Sum_{n >= 1} a(n)*x^n/n! = series reversion( 1/(1 + x)^4*log(1 + x) ) = series reversion(x - 9*x^2/2! + 74*x^3/3! - 638*x^4/4! + ...). Cf. A000272, A052750. - Peter Bala, Jun 15 2016

Better description from Vladeta Jovovic, Sep 02 2003

A052782 a(n) = (5*n+1)^(n-1).

Original entry on oeis.org

1, 1, 11, 256, 9261, 456976, 28629151, 2176782336, 194754273881, 20047612231936, 2334165173090451, 303305489096114176, 43513917611435838661, 6831675453247426400256, 1165087474585497590531111, 214481724045177216015794176, 42391158275216203514294433201

Offset: 0

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

G. C. Greubel, Table of n, a(n) for n = 0..313
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 739
J.-C. Novelli, 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, A052752, A052774, A000272, A001721.

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

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

With[{nmax = 50}, CoefficientList[Series[Exp[-LambertW[-5*x]/5], {x, 0, nmax}], x]*Range[0, nmax]!] (* or *) Table[(5*n+1)^(n-1), {n,0,50}] (* G. C. Greubel, Nov 16 2017 *)

for(n=0,50, print1((5*n+1)^(n-1), ", ")) \\ G. C. Greubel, Nov 16 2017

x='x+O('x^50); Vec(serlaplace(exp(-lambertw(-5*x)/5))) \\ G. C. Greubel, Nov 16 2017

E.g.f.: exp(-1/5*LambertW(-5*x)).
From Peter Bala, Dec 19 2013: (Start)
The e.g.f. A(x) = 1 + x + 11*x^2/2! + 256*x^3/3! + 9261*x^4/4! + ... satisfies:
1) A(x*exp(-5*x)) = exp(x) = 1/A(-x*exp(5*x));
2) A^5(x) = 1/x*series reversion(x*exp(-5*x));
3) A(x^5) = 1/x*series reversion(x*exp(-x^5));
4) A(x) = exp(x*A(x)^5);
5) A(x) = 1/A(-x*A(x)^10). (End)
E.g.f.: (-LambertW(-5*x)/(5*x))^(1/5). - Vaclav Kotesovec, Dec 07 2014
Related to A001721 by Sum_{n >= 1} a(n)*x^n/n! = series reversion( 1/(1 + x)^5*log(1 + x) ) = series reversion(x - 11*x^2/2! + 107*x^3/3! - 1066*x^4/4! + ...). Cf. A000272, A052750. - Peter Bala, Jun 15 2016

Better description from Vladeta Jovovic, Sep 02 2003

A360609 E.g.f. satisfies A(x) = exp(x*A(x)^3) / (1-x).

Original entry on oeis.org

1, 2, 17, 313, 9053, 357941, 17975605, 1095604133, 78570635225, 6482415935449, 604889610870881, 62989604872166897, 7241672622495518773, 911048848278644776949, 124497704904842673086285, 18364053909500922198147421, 2908158473059042016441887025

Offset: 0

Seiichi Manyama, Mar 05 2023

nonn

Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A352410, A360601.
Cf. A052752, A361182.
Cf. A370876.

my(N=20, x='x+O('x^N)); Vec(serlaplace((lambertw(-3*x/(1-x)^3)/(-3*x))^(1/3)))

E.g.f.: (LambertW( -3*x/(1-x)^3 ) / (-3*x))^(1/3).
a(n) ~ 3^(-5/6) * (2^(4/3) + 2*(3 + sqrt(4*exp(1) + 9))^(1/3) * exp(-2/3) - 2^(2/3) * (3 + sqrt(4*exp(1) + 9))^(2/3) * exp(-1/3))^(1/6) * 2^(1/3) * (3 + sqrt(4*exp(1) + 9))^(4/9) * sqrt(4 - 2^(4/3) * (3 + sqrt(4*exp(1) + 9))^(2/3) * exp(-1/3) + 3*2^(2/3) * exp(-2/3) * (3 + sqrt(4*exp(1) + 9))^(1/3)) * n^(n-1) * (12 + 4*sqrt(4*exp(1) + 9))^(n/3) / (exp(7/18 + 5*n/3) * (2 - 2^(1/3) * (3 + sqrt(4*exp(1) + 9))^(2/3) * exp(-1/3) + exp(-2/3) * (12 + 4*sqrt(4*exp(1) + 9))^(1/3))^n * ((3 + sqrt(4*exp(1) + 9))^(2/3) * exp(-1/3) - 2^(2/3))^(3/2) * sqrt(2^(1/3) * (3 + sqrt(4*exp(1) + 9))^(2/3) * exp(-1/3) - 2)). - Vaclav Kotesovec, Mar 06 2023
a(n) = n! * Sum_{k=0..n} (3*k+1)^(k-1) * binomial(n+2*k,n-k)/k!. - Seiichi Manyama, Mar 09 2024

cp's OEIS Frontend

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

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A274265 a(n) = (3*n - 1)^(n-1).

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A052774 a(n) = (4*n+1)^(n-1).

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A052782 a(n) = (5*n+1)^(n-1).

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A360609 E.g.f. satisfies A(x) = exp(x*A(x)^3) / (1-x).

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A384855 E.g.f. A(x) satisfies A(x) = exp( x/A(-x*A(x))^3 ).

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A384856 E.g.f. A(x) satisfies A(x) = exp( x/A(-x*A(x)^2)^3 ).

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