%I #67 Oct 06 2023 10:29:11

%S 1,3,25,343,6561,161051,4826809,170859375,6975757441,322687697779,

%T 16679880978201,952809757913927,59604644775390625,4052555153018976267,

%U 297558232675799463481,23465261991844685929951,1977985201462558877934081,177482997121587371826171875

%N a(n) = (2n+1)^n.

%C a(n) is the determinant of the zigzag matrix Z(n) (see A088961). - _Paul Boddington_, Nov 03 2003

%C 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

%C 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

%C a(n) is the number of rooted (at an edge) 2-trees with n+2 edges. See also A052750. - _Nikos Apostolakis_, Dec 05 2018

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

%H G. C. Greubel, <a href="/A085527/b085527.txt">Table of n, a(n) for n = 0..350</a>

%H Karola Mészáros, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v20i2p31">Labeling the Regions of the Type C_n Shi Arrangement</a>, The Electronic Journal of Combinatorics, vol. 20, no. 2, (2013).

%H Zhi-Wei Sun, Fedor Petrov, <a href="https://mathoverflow.net/questions/321098/">A surprising identity</a>, MathOverflow, Jan 17 2019.

%F E.g.f.: sqrt(2)/(2*(1+LambertW(-2*x))*sqrt(-x/LambertW(-2*x))). - _Vladeta Jovovic_, Oct 16 2004

%F 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

%F a(n) = [x^n] 1/(1 - (2*n+1)*x). - _Ilya Gutkovskiy_, Oct 10 2017

%F 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

%F a(n) ~ exp(1/2) * 2^n * n^n. - _Vaclav Kotesovec_, Dec 05 2021

%F 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

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

%t Table[(2 n + 1)^n, {n, 0, 20}] (* _Wesley Ivan Hurt_, Mar 01 2015 *)

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

%o (Magma) [(2*n+1)^n: n in [0..20]]; // _Wesley Ivan Hurt_, Mar 01 2015

%o (GAP) List([0..20],n->(2*n+1)^n); # _Muniru A Asiru_, Dec 05 2018

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

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_, Jul 05 2003