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%I #11 Jan 15 2019 01:08:34
%S 1,8,35,80,-959,-31176,-434029,-2353376,47273761,2216299240,
%T 54409114979,709671322224,-7010555032415,-769095804780520,
%U -27981768126212461,-591718277116485568,773179823364059329,753403954307904026568,40453080644744591090723,1239578120382146959699600
%N E.g.f.: S(x) = Sum_{n>=0} sin((2*n+1)*x) * x^n / (1 - x^(2*n+1)).
%H Paul D. Hanna, <a href="/A257454/b257454.txt">Table of n, a(n) for n= 1..200</a>
%F E.g.f. S(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) S(x) * (S(x)/R(x))' = - C(x) * (C(x)/R(x))',
%F where
%F (a) R(x) = [ Sum_{n>=0} x^(n*(n+1)) ]^2, and
%F (b) C(x) = Sum_{n>=0} cos((2*n+1)*x) * x^n / (1 + x^(2*n+1)), the e.g.f. of A257453.
%e E.g.f.: S(x) = x + 8*x^2/2! + 35*x^3/3! + 80*x^4/4! - 959*x^5/5! +...
%e where
%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 RELATED SERIES.
%e The dual series
%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 C(x) = 1 + 3*x^2/2! - 24*x^3/3! - 287*x^4/4! - 2480*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, sin((2*m+1)*x +x*O(x^n)) * x^m/(1-x^(2*m+1)) )); n!*polcoeff(A,n)}
%o for(n=1,30,print1(a(n),", "))
%Y Cf. A257453, A257215, A008438.
%K sign
%O 1,2
%A _Paul D. Hanna_, Apr 23 2015
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