A257214 - OEIS

%I #11 Jun 22 2016 10:58:46

%S 1,0,5,24,337,3280,46501,811496,15270977,318449952,7554700261,

%T 194401167928,5484157128913,167431552506608,5496127228989125,

%U 193614639911456520,7265814918674507521,289758831638674507840,12237733598089127162437,545392221565792906192472,25589486575413268343127761,1260584085915542118144276240

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

%H Paul D. Hanna, <a href="/A257214/b257214.txt">Table of n, a(n) for n = 0..200</a>

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

%F (1) C(x)^2 - S(x)^2 = R(x)^2,

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

%F where

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

%F (b) S(x) = Sum_{n>=0} sinh((2*n+1)*x) * x^n / (1 - x^(2*n+1)) the e.g.f. of A257215.

%e E.g.f.: C(x) = 1 + 5*x^2/2! + 24*x^3/3! + 337*x^4/4! + 3280*x^5/5! +...

%e where

%e 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 RELATED SERIES.

%e The dual Lambert series

%e 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 S(x) = x + 8*x^2/2! + 37*x^3/3! + 304*x^4/4! + 4081*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 +...+ A008438(n)*x^(2*n) +...

%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 + 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! +...

%e 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! +...

%e 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 +...

%e The normalized series begin

%e 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! +...

%e 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! +...

%e (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! +...

%e where

%e 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! +...

%e 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))].

%o (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)}

%o for(n=0,30,print1(a(n),", "))

%Y Cf. A257215, A008438.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Apr 18 2015