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

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

KEYWORD

nonn,tabf

AUTHOR

Daniel Tisdale, Sep 20 2008

EXTENSIONS

Edited and more terms from Michel Marcus, Aug 14 2022

STATUS

approved