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

0,2

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

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

Sequence in context: A138438 A320646 A241709 * A192724 A078422 A291897

Adjacent sequences: A085525 A085526 A085527 * A085529 A085530 A085531

Keyword

nonn,easy

Author

N. J. A. Sloane, Jul 05 2003

Status

approved