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

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

Offset: 0

Author

N. J. A. Sloane, Jul 05 2003

Keywords

• nonn
• easy

Comments

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

References

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

Links

• 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.

Crossrefs

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

Programs

GAP

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

Magma

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

Maple

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

Mathematica

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

PARI

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

Python

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

Formula

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