login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

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
OFFSET
1,2
LINKS
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
Sequence in context: A221633 A201452 A128246 * A204076 A319960 A163832
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Apr 18 2015
STATUS
approved