The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A257215 E.g.f.: S(x) = Sum_{n>=0} sinh((2*n+1)*x) * x^n / (1 - x^(2*n+1)). 4
 1, 8, 37, 304, 4081, 51384, 837733, 15583712, 324393985, 7669671400, 195589720261, 5509114219536, 168051665376817, 5506719600441752, 193872344999763781, 7271477485665147328, 289936454250117720193, 12242148798010459653576, 545520427163375125201381, 25593712286164576808576240 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 LINKS Paul D. Hanna, Table of n, a(n) for n = 1..200 FORMULA E.g.f. S(x) satisfies: (1) C(x)^2 - S(x)^2 = R(x)^2, (2) S(x) * (S(x)/R(x))' = C(x) * (C(x)/R(x))', where (a) R(x) = [ Sum_{n>=0} x^(n*(n+1)) ]^2, and (b) C(x) = Sum_{n>=0} cosh((2*n+1)*x) * x^n / (1 + x^(2*n+1)) the e.g.f of A257214. EXAMPLE E.g.f.: S(x) = x + 8*x^2/2! + 37*x^3/3! + 304*x^4/4! + 4081*x^5/5! +... where 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) +... RELATED SERIES. The dual Lambert series 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) +... C(x) = 1 + 5*x^2/2! + 24*x^3/3! + 337*x^4/4! + 3280*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, sinh((2*m+1)*x +x*O(x^n)) * x^m/(1-x^(2*m+1)) )); n!*polcoeff(A, n)} for(n=1, 30, print1(a(n), ", ")) CROSSREFS Cf. A257214, A008438. Sequence in context: A221633 A201452 A128246 * A204076 A319960 A163832 Adjacent sequences:  A257212 A257213 A257214 * A257216 A257217 A257218 KEYWORD nonn AUTHOR Paul D. Hanna, Apr 18 2015 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified September 24 03:26 EDT 2021. Contains 347623 sequences. (Running on oeis4.)