A257216 E.g.f.: Sum_{n>=0} exp((2*n+1)*x) * (1 - x^(2*n+1)*exp(-(4*n+2)*x)) * x^n/(1 - x^(4*n+2)).

1, 1, 13, 61, 641, 7361, 97885, 1649229, 30854689, 642843937, 15224371661, 389990888189, 10993271348449, 335483217883425, 11002846829430877, 387486984911220301, 14537292404339654849, 579695285888792228033, 24479882396099586816013, 1090912648729168031393853, 51183198861577845151704001

Offset

0,3

Links

Table of n, a(n) for n=0..20.

Formula

E.g.f.: C(x) + S(x) where C(x)^2 - S(x)^2 = [ Sum_{n>=0} x^(n*(n+1)) ]^4 and

(a) C(x) = Sum_{n>=0} cosh((2*n+1)*x) * x^n / (1 + x^(2*n+1)),

(b) S(x) = Sum_{n>=0} sinh((2*n+1)*x) * x^n / (1 - x^(2*n+1)).

a(n) = A257214(n) + A257215(n).

Example

 E.g.f.: A(x) = 1 + x + 13*x^2/2! + 61*x^3/3! + 641*x^4/4! + 7361*x^5/5! +...
where A(x) equals the sum of the dual Lambert series
(a) 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) +...
(b) 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) +...
more explicitly,
(a) C(x) = 1 + 5*x^2/2! + 24*x^3/3! + 337*x^4/4! + 3280*x^5/5! +...
(b) S(x) = x + 8*x^2/2! + 37*x^3/3! + 304*x^4/4! + 4081*x^5/5! +...
such that
C(x)^2 - S(x)^2 = [1 + x^2 + x^6 + x^12 + x^20 +...+ x^(n^2+n) +...]^4
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 +...
RELATED SERIES.
A(x)*A(-x) = 1 + 24*x^2/2! + 1808*x^4/4! + 283008*x^6/6! + 85053312*x^8/8! +...
sqrt(A(x)*A(-x)) = 1 + 12*x^2/2! + 472*x^4/4! + 56544*x^6/6! + 15730432*x^8/8! + 7023569920*x^10/10! + 4635702211584*x^12/12! +...

Prog

 (PARI) {a(n)=local(X=x +x*O(x^n), A=1); A = sum(m=0, n, exp((2*m+1)*X) * (1 - x^(2*m+1)*exp(-(4*m+2)*X)) * x^m/(1 - X^(4*m+2))); n!*polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) /* As the sum of dual Lambert series (A257214, A257215): */
{A257214(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)}
{A257215(n)=local(A = sum(m=0, n, sinh((2*m+1)*x +x*O(x^n)) * x^m/(1-x^(2*m+1)) )); n!*polcoeff(A, n)}
{a(n) = A257214(n) + A257215(n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A257214, A257215.

Sequence in context: A279762 A147185 A122885 * A135535 A158870 A145044

Adjacent sequences: A257213 A257214 A257215 * A257217 A257218 A257219

Keyword

nonn

Author

Paul D. Hanna, Apr 19 2015

Status

approved