A326552 E.g.f. S(x), where C(x*y) + iS(x*y) = exp( i*Integral Integral C(x*y) dx dy ) such that C(x)^2 + S(x)^2 = 1.

1, -8, 576, -160768, 123535360, -212713734144, 716196297048064, -4280584942657732608, 42250703121584165486592, -651154631135458759089848320, 14983590319172065236171175755776, -496301942561421311900528265903734784, 22953613919171561374366988621726483480576, -1444609513446024762131466039751756562435145728, 121022534222796916421149671221445519229890299166720

Offset

1,2

Comments

The hyperbolic analog of the e.g.f. is described by A325292.

The e.g.f. can be derived from the functions described by A326797, A326798, and A326799.

The e.g.f. can be derived from the functions described by A326800, A326801, and A326802.

Links

Table of n, a(n) for n=1..15.

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) S'(x)/C(x) = -C'(x)/S(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) + i*S(x) = exp( i*Integral 1/x * (Integral C(x) dx) dx ).

(3.b) C(x) = cos( Integral 1/x * (Integral C(x) dx) dx ).

(3.c) S(x) = sin( 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) + i*S(x*y) = exp( i*Integral Integral C(x*y) dx dy ).

(5.b) C(x*y) = cos( Integral Integral C(x*y) dx dy ).

(5.c) S(x*y) = sin( 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) = sin( Integral 1/x * (Integral C(x) dx) dx ),
also, S(x*y) = sin( Integral Integral C(x*y) dx dy ),
such that C(x)^2 + S(x)^2 = 1.
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) = cos( Integral 1/x * (Integral C(x) dx) dx ),
also, C(x*y) = cos( Integral Integral C(x*y) dx dy ).
RELATED FUNCTIONS.
Given functions Ax, Bx, Cx, Ay, By, and Cy defined by
(1a) Ax = 0 + Integral Bx*Cy - Cx*By dx,
(1b) Bx = 1 + Integral Cx*Ay - Ax*Cy dx,
(1c) Cx = 0 + Integral Ax*By - Bx*Ay dx,
(2a) Ay = 0 + Integral By*Cx - Cy*Bx dy,
(2b) By = 0 + Integral Cy*Ax - Ay*Cx dy,
(2c) Cy = 1 + Integral Ay*Bx - By*Ax dy,
then
S(x*y) = Ax*Ay + Bx*By + Cx*Cy.
These related series begin as follows.
Ax = x + (-1*x^3 - 3*x*y^2)/3! + (1*x^5 - 30*x^3*y^2 + 5*x*y^4)/5! + (-1*x^7 + 315*x^5*y^2 + 525*x^3*y^4 - 7*x*y^6)/7! + (1*x^9 - 1260*x^7*y^2 + 18270*x^5*y^4 - 2940*x^3*y^6 + 9*x*y^8)/9! + (-1*x^11 + 3465*x^9*y^2 - 496650*x^7*y^4 - 695310*x^5*y^6 + 10395*x^3*y^8 - 11*x*y^10)/11! + ... (A326797)
Bx = 1 + (-1*x^2)/2! + (1*x^4)/4! + (-1*x^6 + 120*x^4*y^2)/6! + (1*x^8 - 672*x^6*y^2)/8! + (-1*x^10 + 2160*x^8*y^2 - 120960*x^6*y^4)/10! + (1*x^12 - 5280*x^10*y^2 + 1584000*x^8*y^4)/12! + ... (A326798)
Cx = (2*x*y)/2! + (-4*x*y^3)/4! + (-160*x^3*y^3 + 6*x*y^5)/6! + (1344*x^3*y^5 - 8*x*y^7)/8! + (145152*x^5*y^5 - 5760*x^3*y^7 + 10*x*y^9)/10! + (-2534400*x^5*y^7 + 17600*x^3*y^9 - 12*x*y^11)/12! + ... (A326799)
Ay = -y + (3*x^2*y + 1*y^3)/3! + (-5*x^4*y + 30*x^2*y^3 + -1*y^5)/5! + (7*x^6*y + -525*x^4*y^3 + -315*x^2*y^5 + 1*y^7)/7! + (-9*x^8*y + 2940*x^6*y^3 + -18270*x^4*y^5 + 1260*x^2*y^7 + -1*y^9)/9! + (11*x^10*y + -10395*x^8*y^3 + 695310*x^6*y^5 + 496650*x^4*y^7 + -3465*x^2*y^9 + 1*y^11)/11! + ...
By = (2*x*y)/2! + (-4*x^3*y)/4! + (6*x^5*y + -160*x^3*y^3)/6! + (-8*x^7*y + 1344*x^5*y^3)/8! + (10*x^9*y + -5760*x^7*y^3 + 145152*x^5*y^5)/10! + (-12*x^11*y + 17600*x^9*y^3 + -2534400*x^7*y^5)/12! + ...
Cy = 1 + (-1*y^2)/2! + (1*y^4)/4! + (120*x^2*y^4 + -1*y^6)/6! + (-672*x^2*y^6 + 1*y^8)/8! + (-120960*x^4*y^6 + 2160*x^2*y^8 + -1*y^10)/10! + (1584000*x^4*y^8 + -5280*x^2*y^10 + 1*y^12)/12! + ...

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. A326551, A325292.

Cf. A326797, A326798, A326799.

Cf. A326800, A326801, A326802.

Sequence in context: A086641 A248706 A325292 * A303933 A188780 A015023

Adjacent sequences: A326549 A326550 A326551 * A326553 A326554 A326555

Keyword

sign

Author

Paul D. Hanna, Jul 25 2019

Status

approved