The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A325292 E.g.f. S(x), where C(x*y) + S(x*y) = exp( Integral Integral C(x*y) dx dy ) such that C(x)^2 - S(x)^2 = 1. 4
1, 8, 576, 160768, 123535360, 212713734144, 716196297048064, 4280584942657732608, 42250703121584165486592, 651154631135458759089848320, 14983590319172065236171175755776, 496301942561421311900528265903734784, 22953613919171561374366988621726483480576, 1444609513446024762131466039751756562435145728 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Unsigned version of A326552.
LINKS
FORMULA
E.g.f. S(x) = Sum_{n>=0} a(n)*x^(2*n+1)/(2*n+1)!^2, where series S(x) and related series C(x) satisfy the following relations.
(1.a) C(x)^2 - S(x)^2 = 1.
(1.b) C'(x)/S(x) = S'(x)/C(x) = 1/x * Integral C(x) dx.
(2.a) S(x) = Integral C(x)/x * (Integral C(x) dx) dx.
(2.b) C(x) = 1 + Integral S(x)/x * (Integral C(x) dx) dx.
(3.a) C(x) + S(x) = exp( Integral 1/x * (Integral C(x) dx) dx ).
(3.b) C(x) = cosh( Integral 1/x * (Integral C(x) dx) dx ).
(3.c) S(x) = sinh( Integral 1/x * (Integral C(x) dx) dx ).
Integration.
(4.a) S(x*y) = Integral C(x*y) * (Integral C(x*y) dy) dx.
(4.b) C(x*y) = 1 + Integral S(x*y) * (Integral C(x*y) dy) dx.
(4.c) S(x*y) = Integral C(x*y) * (Integral C(x*y) dx) dy.
(4.d) C(x*y) = 1 + Integral S(x*y) * (Integral C(x*y) dx) dy.
Exponential.
(5.a) C(x*y) + S(x*y) = exp( Integral Integral C(x*y) dx dy ).
(5.b) C(x*y) = cosh( Integral Integral C(x*y) dx dy ).
(5.c) S(x*y) = sinh( Integral Integral C(x*y) dx dy ).
Derivatives.
(6.a) d/dx S(x*y) = C(x*y) * Integral C(x*y) dy.
(6.b) d/dx C(x*y) = S(x*y) * Integral C(x*y) dy.
(6.c) d/dy S(x*y) = C(x*y) * Integral C(x*y) dx.
(6.d) d/dy C(x*y) = S(x*y) * Integral C(x*y) dx.
EXAMPLE
E.g.f. S(x) = x + 8*x^3/3!^2 + 576*x^5/5!^2 + 160768*x^7/7!^2 + 123535360*x^9/9!^2 + 212713734144*x^11/11!^2 + 716196297048064*x^13/13!^2 + 4280584942657732608*x^15/15!^2 + 42250703121584165486592*x^17/17!^2 + 651154631135458759089848320*x^19/19!^2 + 14983590319172065236171175755776*x^21/21!^2 + ...
where S(x) = sinh( Integral 1/x * (Integral C(x) dx) dx ),
also, S(x*y) = sinh( Integral Integral C(x*y) dx dy ).
RELATED SERIES.
C(x) = 1 + 2*x^2/2!^2 + 56*x^4/4!^2 + 8336*x^6/6!^2 + 3985792*x^8/8!^2 + 4679517952*x^10/10!^2 + 11427218287616*x^12/12!^2 + 51793067942397952*x^14/14!^2 + 400951893341645930496*x^16/16!^2 + 4975999084909976839454720*x^18/18!^2 + 94178912073481319162642169856*x^20/20!^2 + ...
where C(x) = cosh( Integral 1/x * (Integral C(x) dx) dx ),
also, C(x*y) = cosh( Integral Integral C(x*y) dx dy ).
SPECIFIC VALUES.
At x = 1/2,
C(1/2) = 1.13133757946411922642102833324416139...
S(1/2) = 0.52907912329606456055608764850290077...
log(C(1/2) + S(1/2)) = 0.50706859662590456104854330721421537...
At x = 1,
C(1) = 1.61616724447561044622618032294959193...
S(1) = 1.26964426597212165112687564431552303...
log(C(1) + S(1)) = 1.05980614652360497313310791544203867...
At x = 2,
C(2) = 7.0181980831554020705059330009720760...
S(2) = 6.9465894030384550946994132182413166...
log(C(2) + S(2)) = 2.636538981679765615420983831302958...
At x = 3, the power series for C(x) and S(x) diverge.
PROG
(PARI) {a(n) = my(C=1, S=x); for(i=1, 2*n+1,
S = intformal( C/x * intformal( C +x*O(x^(2*n+1)) ) );
C = 1 + intformal( S/x * intformal( C +x*O(x^(2*n+1)) ) ); ); (2*n+1)!^2*polcoeff(S, 2*n+1)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
Cf. A325290 (C+S), A325291 (C).
Cf. A326552.
Sequence in context: A058045 A086641 A248706 * A326552 A303933 A188780
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Apr 16 2019
STATUS
approved

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

License Agreements, Terms of Use, Privacy Policy. .

Last modified June 16 07:19 EDT 2024. Contains 373423 sequences. (Running on oeis4.)