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%I #3 Apr 19 2015 08:48:24
%S 1,1,13,61,641,7361,97885,1649229,30854689,642843937,15224371661,
%T 389990888189,10993271348449,335483217883425,11002846829430877,
%U 387486984911220301,14537292404339654849,579695285888792228033,24479882396099586816013,1090912648729168031393853,51183198861577845151704001
%N 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)).
%F E.g.f.: C(x) + S(x) where C(x)^2 - S(x)^2 = [ Sum_{n>=0} x^(n*(n+1)) ]^4 and
%F (a) C(x) = Sum_{n>=0} cosh((2*n+1)*x) * x^n / (1 + x^(2*n+1)),
%F (b) S(x) = Sum_{n>=0} sinh((2*n+1)*x) * x^n / (1 - x^(2*n+1)).
%F a(n) = A257214(n) + A257215(n).
%e E.g.f.: A(x) = 1 + x + 13*x^2/2! + 61*x^3/3! + 641*x^4/4! + 7361*x^5/5! +...
%e where A(x) equals the sum of the dual Lambert series
%e (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) +...
%e (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) +...
%e more explicitly,
%e (a) C(x) = 1 + 5*x^2/2! + 24*x^3/3! + 337*x^4/4! + 3280*x^5/5! +...
%e (b) S(x) = x + 8*x^2/2! + 37*x^3/3! + 304*x^4/4! + 4081*x^5/5! +...
%e such that
%e C(x)^2 - S(x)^2 = [1 + x^2 + x^6 + x^12 + x^20 +...+ x^(n^2+n) +...]^4
%e 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 +...
%e RELATED SERIES.
%e A(x)*A(-x) = 1 + 24*x^2/2! + 1808*x^4/4! + 283008*x^6/6! + 85053312*x^8/8! +...
%e 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! +...
%o (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)}
%o for(n=0,30,print1(a(n),", "))
%o (PARI) /* As the sum of dual Lambert series (A257214, A257215): */
%o {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)}
%o {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)}
%o {a(n) = A257214(n) + A257215(n)}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A257214, A257215.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Apr 19 2015
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