A257214 E.g.f.: C(x) = Sum_{n>=0} cosh((2*n+1)*x) * x^n / (1 + x^(2*n+1)).

1, 0, 5, 24, 337, 3280, 46501, 811496, 15270977, 318449952, 7554700261, 194401167928, 5484157128913, 167431552506608, 5496127228989125, 193614639911456520, 7265814918674507521, 289758831638674507840, 12237733598089127162437, 545392221565792906192472, 25589486575413268343127761, 1260584085915542118144276240

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,

(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} sinh((2*n+1)*x) * x^n / (1 - x^(2*n+1)) the e.g.f. of A257215.

Example

E.g.f.: C(x) = 1 + 5*x^2/2! + 24*x^3/3! + 337*x^4/4! + 3280*x^5/5! +...
where
C(x) = cosh(x)/(1+x) + cosh(3*x)*x/(1+x^3) + cosh(5*x)*x^2/(1+x^5) + cosh(7*x)*x^3/(1+x^7) + cosh(9*x)*x^4/(1+x^9) + cosh(11*x)*x^5/(1+x^11) +...
RELATED SERIES.
The dual Lambert series
S(x) = sinh(x)/(1-x) + sinh(3*x)*x/(1-x^3) + sinh(5*x)*x^2/(1-x^5) + sinh(7*x)*x^3/(1-x^7) + sinh(9*x)*x^4/(1-x^9) + sinh(11*x)*x^5/(1-x^11) +...
S(x) = x + 8*x^2/2! + 37*x^3/3! + 304*x^4/4! + 4081*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 +...+ A008438(n)*x^(2*n) +...
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 + 10*x^2/2! + 48*x^3/3! + 824*x^4/4! + 8960*x^5/5! + 155072*x^6/6! + 2877952*x^7/7! + 60328704*x^8/8! + 1395081216*x^9/9! +...
S(x)^2 = 2*x^2/2! + 48*x^3/3! + 680*x^4/4! + 8960*x^5/5! + 149312*x^6/6! + 2877952*x^7/7! + 59804544*x^8/8! + 1395081216*x^9/9! +...
R(x)^2 = C(x)^2 - S(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 +...
The normalized series begin
C(x)/R(x) = 1 + x^2/2! + 24*x^3/3! + 289*x^4/4! + 2320*x^5/5! + 27361*x^6/6! + 596456*x^7/7! + 11600065*x^8/8! +...
S(x)/R(x) = x + 8*x^2/2! + 25*x^3/3! + 112*x^4/4! + 2961*x^5/5! + 41784*x^6/6! + 557929*x^7/7! + 10393184*x^8/8! +...
(C(x) + S(x))/R(x) = 1 + x + 9*x^2/2! + 49*x^3/3! + 401*x^4/4! + 5281*x^5/5! + 69145*x^6/6! + 1154385*x^7/7! + 21993249*x^8/8! +...
where
C(x) + S(x) = 1 + x + 13*x^2/2! + 61*x^3/3! + 641*x^4/4! + 7361*x^5/5! + 97885*x^6/6! + 1649229*x^7/7! + 30854689*x^8/8! +...
C(x) + S(x) = Sum_{n>=0} [exp((2*n+1)*x)*x^n/(1-x^(4*n+2)) - exp(-(2*n+1)*x)*x^(3*n+1)/(1-x^(4*n+2))].

Prog

(PARI) {a(n)=local(A = sum(m=0, n, cosh((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. A257215, A008438.

Sequence in context: A297664 A051812 A267279 * A003224 A270581 A156310

Adjacent sequences: A257211 A257212 A257213 * A257215 A257216 A257217

Keyword

nonn

Author

Paul D. Hanna, Apr 18 2015

Status

approved