A257453 E.g.f.: C(x) = Sum_{n>=0} cos((2*n+1)*x) * x^n / (1 + x^(2*n+1)).

1, 0, 3, -24, -287, -2480, -6061, 196504, 6666465, 124381728, 1152761219, -16400751928, -1124717924351, -33594921946768, -573356313677421, 3172375291503480, 680727732593163841, 30107084674604991040, 772334689398136241795, 2396611523246866389928, -1018886965683104743074399

Offset

0,3

Links

Paul D. Hanna, Table of n, a(n) for n = 0..200

Formula

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

(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,

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

where

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

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

Example

E.g.f.: C(x) = 1 + 3*x^2/2! - 24*x^3/3! - 287*x^4/4! - 2480*x^5/5! +...
where
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) +...
RELATED SERIES.
The dual series
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) +...
S(x) = x + 8*x^2/2! + 35*x^3/3! + 80*x^4/4! - 959*x^5/5! +...
is related by
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 +...
such that
R(x)^(1/2) = 1 + x^2 + x^6 + x^12 + x^20 + x^30 + x^42 +...+ x^(n^2+n) +...
The squares of these related series begin:
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! +...
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! +...
The normalized series begin
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! +...
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! +...
where (C(x)/R(x))^2 + (S(x)/R(x))^2 = 1.

Prog

(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)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A257454, A257214.

Sequence in context: A355794 A355426 A064037 * A128572 A052592 A059381

Adjacent sequences: A257450 A257451 A257452 * A257454 A257455 A257456

Keyword

sign

Author

Paul D. Hanna, Apr 23 2015

Status

approved