A325290 E.g.f. C(x) + 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.

1, 1, 2, 8, 56, 576, 8336, 160768, 3985792, 123535360, 4679517952, 212713734144, 11427218287616, 716196297048064, 51793067942397952, 4280584942657732608, 400951893341645930496, 42250703121584165486592, 4975999084909976839454720, 651154631135458759089848320, 94178912073481319162642169856, 14983590319172065236171175755776, 2610878440961060713599511173791744

Offset

0,3

Links

Paul D. Hanna, Table of n, a(n) for n = 0..400

Formula

E.g.f. C(x) + S(x), where series C(x) and S(x) are given by

(0.a) C(x) + S(x) = Sum_{n>=0} a(n)*x^n/(n!)^2,

(0.b) C(x) = Sum_{n>=0} a(2*n)*x^(2*n)/(2*n)!^2,

(0.c) S(x) = Sum_{n>=0} a(2*n+1)*x^(2*n+1)/(2*n+1)!^2,

and 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. C(x) + S(x) = 1 + x + 2*x^2/2!^2 + 8*x^3/3!^2 + 56*x^4/4!^2 + 576*x^5/5!^2 + 8336*x^6/6!^2 + 160768*x^7/7!^2 + 3985792*x^8/8!^2 + 123535360*x^9/9!^2 + 4679517952*x^10/10!^2 + 212713734144*x^11/11!^2 + 11427218287616*x^12/12!^2 + 716196297048064*x^13/13!^2 + 51793067942397952*x^14/14!^2 + 4280584942657732608*x^15/15!^2 + 400951893341645930496*x^16/16!^2 + 42250703121584165486592*x^17/17!^2 + 4975999084909976839454720*x^18/18!^2 + 651154631135458759089848320*x^19/19!^2 + 94178912073481319162642169856*x^20/20!^2 + ...
where C(x*y) + S(x*y) = exp( Integral Integral C(x*y) dx dy )
such that C(x)^2 - S(x)^2 = 1.
The e.g.f. as a series of reduced fractional coefficients begins
C(x) + S(x) = 1 + x + 1/2*x^2 + 2/9*x^3 + 7/72*x^4 + 1/25*x^5 + 521/32400*x^6 + 628/99225*x^7 + 31139/12700800*x^8 + 1508/1607445*x^9 + 18279367/51438240000*x^10 + 1081918/8104201875*x^11 + 11159392859/224064973440000*x^12 + 97574002/5282781879375*x^13 + 25289583956249/3710964090113280000*x^14 + 37798925176/15099951538546875*x^15 + 4078693576473449/4453156908135936000000*x^16 + ...
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 ).
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 ).
The sum at x = 1 evaluates to
Sum_{n>=0} a(n)/n!^2 = 2.885811510447732097353055967265114966697682979695060754...
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, n,
S = intformal( C/x * intformal( C +x*O(x^n) ) );
C = 1 + intformal( S/x * intformal( C +x*O(x^n) ) ); ); n!^2*polcoeff(C+S, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A325291 (C), A325292 (S).

Sequence in context: A326009 A372160 A349562 * A197949 A363589 A243953

Adjacent sequences: A325287 A325288 A325289 * A325291 A325292 A325293

Keyword

nonn

Author

Paul D. Hanna, Apr 16 2019

Status

approved