A144732 Triangle of numerator coefficients, reading across rows, of Sum_{k=1..n} (1/(1 + r^2 - 2*r*cos(k*Pi/n))).

1, 2, 2, 2, 3, 4, 5, 4, 3, 4, 6, 8, 8, 8, 6, 4, 5, 8, 11, 12, 13, 12, 11, 8, 5, 6, 10, 14, 16, 18, 18, 18, 16, 14, 10, 6, 7, 12, 17, 20, 23, 24, 25, 24, 23, 20, 17, 12, 7, 8, 14, 20, 24, 28, 30, 32, 32, 32, 30, 28, 24, 20, 14, 8, 9, 16, 23, 28, 33, 36, 39, 40, 41, 40, 39, 36, 33, 28, 23, 16, 9

Offset

1,2

Comments

Diagonals of the triangle of numerators have differences of 1, then 2, then 3, etc.

The limit as n -> infinity of the ratio of polynomials is 1/(1-r^2), 0 < r < 1, which is proved at the Mathematics Stack Exchange link below.

Links

Table of n, a(n) for n=1..81.

Mathematics Stack Exchange, Limit of sequence of rational polynomials.

Eric Weisstein's World of Mathematics, Poisson Integral.

Example

n=1: 1/(1 + r + r^2).
n=2: (2 + 2*r + 2*r^2)/(2*(1 + 2*r + 2*r^2 + 2*r^3 + r^4)).
n=5: (5 + 8r + 11r^2 + 12r^3 + 13r^4 + 12r^5 + 11r^6 + 8r^7 + 5r^8)/(5*(1 + 2r + 2r^2 + 2r^3 + 2r^4 + 2r^5 + 2r^6 + 2r^7 + 2r^8 + 2r^9 + r^10)).
Triangle begins:
  [1]
  [2, 2, 2]
  [3, 4, 5, 4, 3]
  [4, 6, 8, 8, 8, 6, 4]
  [5, 8, 11, 12, 13, 12, 11, 8, 5]
  [6, 10, 14, 16, 18, 18, 18, 16, 14, 10, 6]
  ...

Mathematica

Sum[1/(1+r^2-2rCos[Pi*k/n]), {k, 1, n}]

Prog

(PARI) row(n) = apply(round, Vec(numerator(sum(k=1, n, 1/(1 + x^2 - 2*x*cos(k*Pi/n)))))); \\ Michel Marcus, Aug 14 2022

Crossrefs

Sequence in context: A303846 A275446 A306271 * A055224 A126295 A266686

Adjacent sequences: A144729 A144730 A144731 * A144733 A144734 A144735

Keyword

nonn,tabf

Author

Daniel Tisdale, Sep 20 2008

Extensions

Edited and more terms from Michel Marcus, Aug 14 2022

Status

approved