OFFSET
0,2
COMMENTS
a(n) is the number of polynomials of degree at most n with integer coefficients all having absolute value <= n.
a(n-1) is the number of nodes in the canonical automaton for the affine Weyl group of type D_n. - Tom Edgar, May 12 2016
REFERENCES
Anders Björner and Francesco Brenti, Combinatorics of Coxeter groups. Graduate Texts in Mathematics, 231. Springer, New York, 2005.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..100
FORMULA
From Peter Bala, Aug 06 2012: (Start)
E.g.f.: d/dx{(2*x/T(2*x))^(1/2)*1/(1 - T(2*x))} = 1 + 9*x + 125*x^2/2! + ..., where T(x) is the tree function sum {n >= 1} n^(n-1)*x^n/n! of A000169.
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)^(n-1)*z^(2*n+1)/(2^n*n!). See A214406 for details. In the present case, r = 1, and the resulting e.g.f. is 1/z*U(z)*(1 + U(z)^2 )/(1 - U(z)^2)^3 taken at z = sqrt(2*x).
(End)
Sum_{n>=0} (-1)^n/a(n) = A253299. - Amiram Eldar, Jun 25 2021
MAPLE
seq((2*n+1)^(n+1), n=0..20); # G. C. Greubel, Sep 03 2019
MATHEMATICA
Table[(2*n+1)^(n+1), {n, 0, 20}] (* Vladimir Joseph Stephan Orlovsky, Sep 05 2009, modified by G. C. Greubel, Sep 03 2019 *)
PROG
(Magma) [(2*n+1)^(n+1): n in [0..20]]; // Vincenzo Librandi, May 04 2011
(PARI) vector(20, n, (2*n-1)^n) \\ G. C. Greubel, Sep 03 2019
(Sage) [(2*n+1)^(n+1) for n in (0..20)] # G. C. Greubel, Sep 03 2019
(GAP) List([0..20], n-> (2*n+1)^(n+1)); # G. C. Greubel, Sep 03 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Jul 05 2003
STATUS
approved