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A085527 a(n) = (2n+1)^n. 13
1, 3, 25, 343, 6561, 161051, 4826809, 170859375, 6975757441, 322687697779, 16679880978201, 952809757913927, 59604644775390625, 4052555153018976267, 297558232675799463481, 23465261991844685929951, 1977985201462558877934081, 177482997121587371826171875 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
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
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.
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
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 *)
PROG
(PARI) a(n)=(2*n+1)^n;
(Magma) [(2*n+1)^n: n in [0..20]]; // Wesley Ivan Hurt, Mar 01 2015
(GAP) List([0..20], n->(2*n+1)^n); # Muniru A Asiru, Dec 05 2018
CROSSREFS
Sequence in context: A154961 A322760 A325286 * A093360 A161629 A129506
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Jul 05 2003
STATUS
approved

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Last modified March 29 02:16 EDT 2024. Contains 371264 sequences. (Running on oeis4.)