

A085527


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


10



1, 3, 25, 343, 6561, 161051, 4826809, 170859375, 6975757441, 322687697779, 16679880978201, 952809757913927, 59604644775390625, 4052555153018976267, 297558232675799463481, 23465261991844685929951, 1977985201462558877934081, 177482997121587371826171875
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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 rholabeled graphs with n edges. A graph with n edges is a rholabeled graph if there exists a onetoone mapping from its vertex set to {0,1,...,2n} such that every edge receives as a label the absolute difference of its endvertices and the edge labels are x1,x2,...,xn where xi=i or xi=2n+1i.  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) 2trees 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).


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 (2*n+1)^(n+r) can be expressed in terms of the function U(z) = sum {n >= 0} (2*n+1)^(n1)*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


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

Cf. A062971, A085528, A088961, A099753, A214406, A052750.
Sequence in context: A154961 A322760 A325286 * A093360 A161629 A129506
Adjacent sequences: A085524 A085525 A085526 * A085528 A085529 A085530


KEYWORD

nonn,easy


AUTHOR

N. J. A. Sloane, Jul 05 2003


STATUS

approved



