A144732 - OEIS

%I #38 Aug 16 2022 15:30:49

%S 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,

%T 18,18,16,14,10,6,7,12,17,20,23,24,25,24,23,20,17,12,7,8,14,20,24,28,

%U 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

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

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

%C 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.

%H Mathematics Stack Exchange, <a href="http://math.stackexchange.com/questions/77072">Limit of sequence of rational polynomials</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PoissonIntegral.html">Poisson Integral</a>.

%e n=1: 1/(1 + r + r^2).

%e n=2: (2 + 2*r + 2*r^2)/(2*(1 + 2*r + 2*r^2 + 2*r^3 + r^4)).

%e 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)).

%e Triangle begins:

%e   [1]

%e   [2, 2, 2]

%e   [3, 4, 5, 4, 3]

%e   [4, 6, 8, 8, 8, 6, 4]

%e   [5, 8, 11, 12, 13, 12, 11, 8, 5]

%e   [6, 10, 14, 16, 18, 18, 18, 16, 14, 10, 6]

%e   ...

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

%o (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

%K nonn,tabf

%O 1,2

%A _Daniel Tisdale_, Sep 20 2008

%E Edited and more terms from _Michel Marcus_, Aug 14 2022