A179897 a(n) = (n^(2*n+1) + 1) / (n+1).

1, 1, 11, 547, 52429, 8138021, 1865813431, 593445188743, 250199979298361, 135085171767299209, 90909090909090909091, 74619186937936447687211, 73381705110822317661638341, 85180949465178001182799643437, 115244915978498073437814463065839, 179766618030828831251710653305053711

Offset: 0

Author

Martin Saturka (martin(AT)saturka.net), Jul 31 2010

Keywords

• easy
• nonn

Comments

a(n) is the arithmetic mean of the multiset consisting of n lots of 1/n and one lot of n^(2*n+1). This multiset also has an integer valued geometric mean which is equal to n for n > 0.

According to search at OEIS for particular sequence members, a(n) is also: (1+2*n)-th q-integer for q=-n, (2*(n+1))-th cyclotomic polynomial at q=-n, Gaussian binomial coefficient [2*n+1, 2*n] for q=-n, number of walks of length 1+2*n between any two distinct vertices of the complete graph K_(n+1).

Examples

			For n = 2, a(2) = 11 which is the arithmetic mean of {1/2, 1/2, 2^5} = 33 / 3 = 11. The geometric mean is 8^(1/3) = 2, i.e. both are integral.
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..100
• Google Groups, Integer-valued arithmetic and geometric means of sequences with non-integer numbers

Crossrefs

Main diagonal of A362783.

Values for n = 5, 6 via other ways. Q-integers: A014986, A014987, K_n paths: A015531, A015540, Cyclotomic polynomials: A020504, A020505, Gaussian binomial coefficients: A015391, A015429.

Programs

PARI

a(n) = (n^(2*n + 1) + 1)/(n + 1) \\ Andrew Howroyd, May 03 2023

Python

[(n**(2*n+1)+1)//(n+1) for n in range(1,11)]

Formula

a(n) = Sum_{i=0..2*n} (-n)^i.

Extensions

Edited, a(0)=1 prepended and more terms from Andrew Howroyd, May 03 2023