A085528 a(n) = (2*n+1)^(n+1).

1, 9, 125, 2401, 59049, 1771561, 62748517, 2562890625, 118587876497, 6131066257801, 350277500542221, 21914624432020321, 1490116119384765625, 109418989131512359209, 8629188747598184440949, 727423121747185263828481, 65273511648264442971824673

Offset: 0

Author

N. J. A. Sloane, Jul 05 2003

Keywords

• nonn
• easy

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

Crossrefs

Cf. A000169, A085527, A099753, A214406, A253299.

Programs

GAP

List([0..20], n-> (2*n+1)^(n+1)); # G. C. Greubel, Sep 03 2019

Magma

[(2*n+1)^(n+1): n in [0..20]]; // Vincenzo Librandi, May 04 2011

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 *)

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

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