A257453 - OEIS

%I #10 Jun 22 2016 10:58:55

%S 1,0,3,-24,-287,-2480,-6061,196504,6666465,124381728,1152761219,

%T -16400751928,-1124717924351,-33594921946768,-573356313677421,

%U 3172375291503480,680727732593163841,30107084674604991040,772334689398136241795,2396611523246866389928,-1018886965683104743074399

%N E.g.f.: C(x) = Sum_{n>=0} cos((2*n+1)*x) * x^n / (1 + x^(2*n+1)).

%H Paul D. Hanna, <a href="/A257453/b257453.txt">Table of n, a(n) for n = 0..200</a>

%F E.g.f. C(x) satisfies:

%F (1) C(x)^2 + S(x)^2 = R(x)^2, which is an o.g.f. of A008438, the sum of divisors of the positive odd numbers,

%F (2) C(x) * (C(x)/R(x))' = - S(x) * (S(x)/R(x))',

%F where

%F (a) R(x) = [ Sum_{n>=0} x^(n*(n+1)) ]^2, and

%F (b) S(x) = Sum_{n>=0} sin((2*n+1)*x) * x^n / (1 - x^(2*n+1)), the e.g.f. of A257454.

%e E.g.f.: C(x) = 1 + 3*x^2/2! - 24*x^3/3! - 287*x^4/4! - 2480*x^5/5! +...

%e where

%e C(x) = cos(x)/(1+x) + cos(3*x)*x/(1+x^3) + cos(5*x)*x^2/(1+x^5) + cos(7*x)*x^3/(1+x^7) + cos(9*x)*x^4/(1+x^9) + cos(11*x)*x^5/(1+x^11) +...

%e RELATED SERIES.

%e The dual series

%e S(x) = sin(x)/(1-x) + sin(3*x)*x/(1-x^3) + sin(5*x)*x^2/(1-x^5) + sin(7*x)*x^3/(1-x^7) + sin(9*x)*x^4/(1-x^9) + sin(11*x)*x^5/(1-x^11) +...

%e S(x) = x + 8*x^2/2! + 35*x^3/3! + 80*x^4/4! - 959*x^5/5! +...

%e is related by

%e C(x)^2 + S(x)^2 = R(x)^2 = 1 + 4*x^2 + 6*x^4 + 8*x^6 + 13*x^8 + 12*x^10 + 14*x^12 + 24*x^14 + 18*x^16 + 20*x^18 + 32*x^20 +...

%e such that

%e R(x)^(1/2) = 1 + x^2 + x^6 + x^12 + x^20 + x^30 + x^42 +...+ x^(n^2+n) +...

%e The squares of these related series begin:

%e C(x)^2 = 1 + 6*x^2/2! - 48*x^3/3! - 520*x^4/4! - 6400*x^5/5! - 26432*x^6/6! + 562688*x^7/7! + 24746752*x^8/8! +...

%e S(x)^2 = 2*x^2/2! + 48*x^3/3! + 664*x^4/4! + 6400*x^5/5! + 32192*x^6/6! - 562688*x^7/7! - 24222592*x^8/8! +...

%e The normalized series begin

%e C(x)/R(x) = 1 - x^2/2! - 24*x^3/3! - 287*x^4/4! - 1520*x^5/5! + 10079*x^6/6! + 344344*x^7/7! + 5979457*x^8/8! +...

%e S(x)/R(x) = x + 8*x^2/2! + 23*x^3/3! - 112*x^4/4! - 1999*x^5/5! - 27336*x^6/6! - 295513*x^7/7! + 573856*x^8/8! +...

%e where (C(x)/R(x))^2 + (S(x)/R(x))^2 = 1.

%o (PARI) {a(n)=local(A = sum(m=0,n, cos((2*m+1)*x +x*O(x^n)) * x^m/(1+x^(2*m+1)) )); n!*polcoeff(A,n)}

%o for(n=0,30,print1(a(n),", "))

%Y Cf. A257454, A257214.

%K sign

%O 0,3

%A _Paul D. Hanna_, Apr 23 2015